\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 76, pp. 1--32.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/76\hfil Variational methods for a resonant problem]
{Variational methods for a resonant problem\\
  with the $p$-Laplacian in $\mathbb{R}^N$}

\author[B.\ Alziary, J.\ Fleckinger, P.\ Tak\'a\v{c}\hfil EJDE-2004/76\hfilneg]
{B\'en\'edicte Alziary, Jacqueline Fleckinger,  Peter Tak\'a\v{c}}
% in alphabetical order


\address{B\'en\'edicte Alziary \hfill\break
CEREMATH \& UMR MIP\\
Universit\'e Toulouse~1 -- Sciences Sociales\\
21 all\'ees de Brienne, F--31042 Toulouse Cedex, France}
\email{alziary@univ-tlse1.fr}

\address{Jacqueline Fleckinger \hfill\break
CEREMATH \& UMR MIP \\
Universit\'e Toulouse~1 -- Sciences Sociales\\
21 all\'ees de Brienne, F--31042 Toulouse Cedex, France}
\email{jfleck@univ-tlse1.fr}

\address{Peter Tak\'a\v{c} \hfill\break
Fachbereich Mathematik\\
Universit\"at Rostock\\
Universit\"atsplatz 1, D--18055 Rostock, Germany}
\email{peter.takac@mathematik.uni-rostock.de}


\date{}
\thanks{Submitted March 19, 2004. Published May 26, 2004.}
\subjclass[2000]{35P30, 35J20, 47J10, 47J30}
\keywords{$p$-Laplacian, degenerate quasilinear Cauchy problem, \hfill\break\indent
 Fredholm alternative, $(p-1)$-homogeneous problem at resonance,
 saddle point geometry, \hfill\break\indent
 improved Poincar\'e inequality,  second-order Taylor formula}

\begin{abstract}
 The solvability of the resonant Cauchy problem
 $$
  - \Delta_p u = \lambda_1 m(|x|) |u|^{p-2} u + f(x)
    \quad\hbox{in } \mathbb{R}^N ;\quad u\in D^{1,p}(\mathbb{R}^N),
 $$
 in the entire Euclidean space $\mathbb{R}^N$ ($N\geq 1$) is
 investigated as a part of the Fredholm alternative at the first
 (smallest) eigenvalue $\lambda_1$ of the positive $p$-Laplacian
 $-\Delta_p$ on $D^{1,p}(\mathbb{R}^N)$ relative to the weight
 $m(|x|)$. Here, $\Delta_p$ stands for the $p$-Laplacian,
 $m\colon \mathbb{R}_+\to \mathbb{R}_+$ is a weight
 function assumed to be radially symmetric,
 $m\not\equiv 0$ in $\mathbb{R}_+$, and
 $f\colon \mathbb{R}^N\to \mathbb{R}$ is a
 given function satisfying a suitable integrability condition.
 The weight $m(r)$ is assumed to be bounded and to decay fast enough
 as $r\to +\infty$. Let $\varphi_1$ denote the (positive)
 eigenfunction associated with the (simple) eigenvalue $\lambda_1$
 of $-\Delta_p$. If $\int_{\mathbb{R}^N} f\varphi_1 \,{\rm d}x =0$,
 we show that problem has at least one solution $u$
 in the completion $D^{1,p}(\mathbb{R}^N)$ of
 $C_{\mathrm{c}}^1(\mathbb{R}^N)$ endowed with the norm
 $(\int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x)^{1/p}$.
 To establish this existence result, we employ a saddle point method
 if $1 < p < 2$, and an improved Poincar\'e inequality if
 $2\leq p< N$. We use weighted Lebesgue and Sobolev spaces with
 weights depending on $\varphi_1$. The asymptotic behavior of
 $\varphi_1(x)= \varphi_1(|x|)$ as $|x|\to \infty$ plays a crucial role.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}

\newcommand{\eqdef}{\stackrel{{\mathrm {def}}}{=}}


\section{Introduction} \label{s:Intro}

Spectral problems involving quasilinear degenerate or singular
elliptic operators have been an interesting subject of investigation
for quite some time; see e.g.\
{\sc Dr\'abek}~\cite{Drabek-1}
or
{\sc Fu\v{c}\'{\i}k} et al.~\cite{FucikNSS}.
In our present work we focus our attention on the solvability of
the Cauchy problem
%
\begin{equation}
  - \Delta_p u = \lambda\, m(x)\, |u|^{p-2} u + f(x)
    \;\mbox{ in } \mathbb{R}^N ;\qquad u\in D^{1,p}(\mathbb{R}^N) ,
\label{e:BVP.l}
\end{equation}
%
in the entire Euclidean space $\mathbb{R}^N$ ($N\geq 1$).
Here,
$\Delta_p$ stands for the $p$-Laplacian defined by
$\Delta_p u\equiv \mathop{\rm div} ( |\nabla u|^{p-2} \nabla u )$, $1<p<N$,
$\lambda\in \mathbb{R}$ is the spectral parameter,
$m\colon \mathbb{R}^N\to \mathbb{R}_+$ is a weight function assumed to be
radially symmetric, $m\not\equiv 0$ in $\mathbb{R}^N$, and
$f\colon \mathbb{R}^N\to \mathbb{R}$ is a given function satisfying
a suitable integrability condition.
We look for a weak solution to problem \eqref{e:BVP.l}
in the Sobolev space $D^{1,p}(\mathbb{R}^N)$
defined to be the completion of $C_{\mathrm{c}}^1(\mathbb{R}^N)$
under the Sobolev norm
%
\[
 \| u\|_{ D^{1,p}(\mathbb{R}^N) } \eqdef
  \Big( \int_{\mathbb{R}^N} |\nabla u(x)|^p \,{\rm d}x \Big)^{1/p} .
\]
%

If the weight $m(x)$ is measurable, bounded and decays
at least as fast as
$|x|^{- p - \delta}$ as $|x|\to \infty$, with some $\delta > 0$,
the Sobolev imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^p(\mathbb{R}^N; m)$
turns out to be compact, where
$L^p(\mathbb{R}^N; m)$ denotes
the weighted Lebesgue space of all measurable functions
$u\colon \mathbb{R}^N\to \mathbb{R}$ with the norm
%
\[
 \| u\|_{ L^{p}(\mathbb{R}^N; m) } \eqdef
  \Big( \int_{\mathbb{R}^N} |u(x)|^p\, m(x) \,{\rm d}x \Big)^{1/p}
  < \infty .
\]
%
Hence, the Rayleigh quotient
%
\begin{equation}
 \lambda_1\eqdef \inf \Big\{
    \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
  \colon u\in D^{1,p}(\mathbb{R}^N) \;\mbox{ with }\;
    \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x = 1 \Big\}
\label{def.lam_1}
\end{equation}
%
is positive and gives the first (smallest) eigenvalue
$\lambda_1$ of $-\Delta_p$ relative to the weight $m$.
Now take $f$ from the dual space
$D^{-1,p'}(\mathbb{R}^N)$ of $D^{1,p}(\mathbb{R}^N)$, $p'= p/(p-1)$,
with respect to the standard duality
$\langle \,\cdot\, ,\,\cdot\, \rangle$
induced by the inner product on $L^2(\mathbb{R}^N)$.
If $-\infty < \lambda < \lambda_1$ then
the energy functional corresponding to equation \eqref{e:BVP.l},
%
\begin{equation}
  \mathcal{J}_{\lambda}(u) \eqdef
    \frac{1}{p} \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
  - \frac{\lambda}{p} \int_{\mathbb{R}^N} |u|^p\, m(x) \,{\rm d}x
  - \int_{\mathbb{R}^N} f(x) u \,{\rm d}x
\label{def.jl}
\end{equation}
%
defined for $u\in D^{1,p}(\mathbb{R}^N)$,
is weakly lower semicontinuous and coercive on $D^{1,p}(\mathbb{R}^N)$.
Thus, $\mathcal{J}_{\lambda}$ possesses a global minimizer
which provides a weak solution to equation \eqref{e:BVP.l}.

The critical case $\lambda = \lambda_1$
is much more complicated when $p\not= 2$ because
the linear Fredholm alternative cannot be applied.
First, one has to have sufficient information on
the first eigenvalue $\lambda_1$;
we refer the reader to
{\sc Fleckinger} et al.~\cite[Sect.\ 2 and~3]{FMST} or
{\sc Stavrakakis} and {\sc de~Th\'elin} \cite{StavrThelin}.
One has
%
\begin{equation}
  - \Delta_p \varphi_1
  = \lambda_1\, m(x)\, |\varphi_1|^{p-2} \varphi_1
    \;\mbox{ in } \mathbb{R}^N ;\qquad
    \varphi_1\in D^{1,p}(\mathbb{R}^N)\setminus \{ 0\} ,
\label{e:varphi.l_1}
\end{equation}
%
and the eigenvalue $\lambda_1$ is simple,
by a result due to
{\sc Anane} \cite[Th\'eor\`eme~1, p.~727]{Anane-1}
and later generalized by
{\sc Lindqvist} \cite[Theorem 1.3, p.~157]{Lindqvist}.
Moreover, the corresponding eigenfunction $\varphi_1$
can be normalized by
$\|\varphi_1\|_{ L^p(\mathbb{R}^N; m) } = 1$ and
$\varphi_1 > 0$ in $\mathbb{R}^N$,
owing to the strong maximum principle
\cite[Prop.\ 3.2.1 and 3.2.2, p.~801]{Tolksdorf-1} or
\cite[Theorem~5, p.~200]{Vazquez}.
We decompose the unknown function $u\in D^{1,p}(\mathbb{R}^N)$
as a direct sum
%
\begin{equation} \label{ortho:u}
\begin{gathered}
 u = u^\parallel\cdot \varphi_1 + u^\top\quad \mbox{ where }\\
u^\parallel = \int_{\mathbb{R}^N} u\, \varphi_1\, \mu(x) \,{\rm d}x
\in \mathbb{R}     \;\mbox{ and }\;
\int_{\mathbb{R}^N} u^\top\, \varphi_1\, \mu(x) \,{\rm d}x = 0 ,
\end{gathered}
\end{equation}
%
with the weight $\mu(x)$ given by $\mu\eqdef \varphi_1^{p-2}\, m$.
It is quite natural that we treat the two components,
$u^\parallel$ and $u^\top$, differently.
The linearization of the equation
%
\begin{equation}
  - \Delta_p u = \lambda_1\, m(x)\, |u|^{p-2} u + f(x)
    \;\mbox{ in } \mathbb{R}^N ;\qquad u\in D^{1,p}(\mathbb{R}^N) ,
\label{e:BVP.l_1}
\end{equation}
%
about $u^\parallel\cdot \varphi_1$,
and the corresponding ``quadratization'' of the functional
$\mathcal{J}_{\lambda_1}$,
play an important role in our approach.
We will also see that the orthogonality condition
%
\begin{equation}
 \int_{\mathbb{R}^N} f\, \varphi_1\, \mu \,{\rm d}x \equiv
  \int_{\mathbb{R}^N} f\, \varphi_1^{p-1}\, m \,{\rm d}x = 0
\label{e:f.phi_1=0}
\end{equation}
%
for $f$ and $\varphi_1$ relative to the measure
$\mu(x) \,{\rm d}x$
is sufficient, but not necessary for the solvability of
problem~\eqref{e:BVP.l_1}.

Similarly as in
{\sc Dr\'abek} and {\sc Holubov\'a} \cite{DrabHolub}
for $1<p<2$, in
{\sc Fleckinger} and {\sc Tak\'a\v{c}} \cite{FleckTakac-1}
for $2\leq p < \infty$, and in
{\sc Tak\'a\v{c}} \cite{Takac-1, Takac-2}
for any $1 < p < \infty$,
where the domain $\Omega\subset \mathbb{R}^N$ is bounded,
we apply the calculus of variations using the direct sum
\eqref{ortho:u} in order to obtain a solution to
equation~\eqref{e:BVP.l_1}.
We use entirely different variational methods to treat
the two cases $1<p<2$ and $2\leq p < N$:
In the former case we apply a saddle point method from
\cite{DrabHolub, Takac-1, Takac-2},
whereas in the latter case we use a minimization method due to
\cite{FleckTakac-1}
which is based on an improved Poincar\'e inequality.
Our variational methods are different from the standard ones
because the functional $\mathcal{J}_{\lambda_1}$
needs not satisfy the Palais\--Smale condition if
$f$ obeys the orthogonality condition~\eqref{e:f.phi_1=0}; cf.\
{\sc del~Pino}, {\sc Dr\'abek} and {\sc Man\'a\-se\-vich}
\cite[Theorem 1.2(ii), p.~390]{PinoDrabMan}.

This paper is organized as follows.
In Section~\ref{s:Prelims}
we mention some elementary properties of the first eigenfunction
$\varphi_1$ and introduce basic function spaces and notation.
Section~\ref{s:Main}
contains our main results on the solvability of
problem~\eqref{e:BVP.l_1},
Theorem~\ref{thm-Exist:p>2} for $2\leq p < N$ and
Theorem~\ref{thm-Exist:p<2} for $1 < p < 2\leq N$, and
some properties of the energy functional $\mathcal{J}_{\lambda}$
needed to establish the solvability, as well.
Naturally, our approach requires the compactness of
several Sobolev imbeddings in $\mathbb{R}^N$ with weights
(Proposition \ref{prop-Compact}) which we prove in
Section~\ref{s:pf-Compact}.
In Section~\ref{s:Auxiliary} we establish a few auxiliary results for
the quadratization of $\mathcal{J}_{\lambda_1}$.
We use this quadratization to verify
the improved Poincar\'e inequality (Lemma~\ref{lem-Poincare})
for $2\leq p < N$ in Section~\ref{s:Impr_Poinc}.
 From this inequality we derive Theorem~\ref{thm-Exist:p>2}
in Section~\ref{s:pr-Exist:p>2}.
For $1<p<2$ the quadratization of $\mathcal{J}_{\lambda_1}$
is employed in a saddle point method to prove
Theorem~\ref{thm-Exist:p<2} in Section~\ref{s:pr-Exist:p<2}.
Finally, some asymptotic formulas for the eigenfunction
$\varphi_1$ near infinity are established in the Appendix
(Proposition \ref{prop-Asympt}).

The rate of decay of $\varphi_1(x)$ as $|x|\to \infty$ is, in fact,
the main cause for our restriction $p<N$.
The case $p\geq N$ seems to require a different technique.


\section{Preliminaries}
\label{s:Prelims}

We now put our resonant problem \eqref{e:BVP.l_1}
into a rigorous setting.
Set $\mathbb{R}_+ = [0,\infty)$.
For $x\in \mathbb{R}^N$ we denote by $r = |x|\geq 0$
the radial variable in $\mathbb{R}^N$.

\subsection{Hypotheses}
\label{ss:Hypos}
We assume $1<p<N$ throughout this article unless indicated otherwise.
Furthermore, the weight $m$ is assumed to be radially symmetric,
$m(x)\equiv m(|x|)$, $x\in \mathbb{R}^N$, where
$m\colon \mathbb{R}_+\to \mathbb{R}$
is a Lebesgue measurable function satisfying
the following hypothesis:
\begin{enumerate}
\renewcommand{\labelenumi}{({\bf H})}

\item
\makeatletter
\def\@currentlabel{H}\label{hyp:m>0}
\makeatother
There exist constants $\delta > 0$ and $C>0$ such that
%
\begin{equation}
  0 < m(r)\leq \frac{C}{ (1+r)^{p+\delta} }
  \quad\mbox{for almost all } 0\leq r < \infty .
\label{ineq:m>0}
\end{equation}
\end{enumerate}

\begin{remark}\label{rem-hyp:m>0}\begingroup\rm
In fact, in hypothesis \eqref{hyp:m>0} above, instead of
$m(r) > 0$ for almost all $0\leq r < \infty$,
it suffices to assume only $m\geq 0$ a.e.\ in $\mathbb{R}^N$ and
$m$ does not vanish identically near zero, i.e.,
for every $r_0 > 0$ we have $m\not\equiv 0$ in $(0,r_0)$.
However, if
$m\equiv 0$ on a set $S\subset \mathbb{R}_+$ of positive Lebesgue measure,
then the weighted spaces
$\mathcal{H}_{\varphi_1} = L^2(\mathbb{R}^N; \varphi_1^{p-2} m)$,
$L^p(\mathbb{R}^N; m)$, etc.\ defined below
become linear spaces with a seminorm only.
Moreover, all functions from their dual spaces
$  \mathcal{H}_{\varphi_1}' =
  L^{2}\big( \mathbb{R}^N; \varphi_1^{2-p} m^{-1} \big)$,
$L^{p'}( \mathbb{R}^N; m^{-1/(p-1)} )$, etc., respectively,
must vanish identically (i.e., almost everywhere)
in the ``spherical shell'' $\{ x\in \mathbb{R}^N\colon |x|\in S\}$.
This would make our presentation much less clear; therefore,
we have decided to leave the necessary amendments in our arguments to
an interested reader.
\endgroup
\end{remark}


\subsection{The first eigenfunction $\varphi_1$}
\label{ss:phi_1}

Under hypothesis \eqref{hyp:m>0},
the first eigenvalue $\lambda_1$ of $-\Delta_p$ on $\mathbb{R}^N$
relative to the weight $m(|x|)$ is simple and
the eigenfunction $\varphi_1$ associated with $\lambda_1$
is commonly called
a ``ground state'' for the Cauchy problem~\eqref{e:varphi.l_1}.
The simplicity of $\lambda_1$ forces
$\varphi_1(x) = \varphi_1(|x|)$ radially symmetric in $\mathbb{R}^N$.
Hence, the eigenvalue problem \eqref{e:varphi.l_1} is equivalent to
%
\begin{equation*}
\begin{split}
{}- ( |\varphi_1'|^{p-2} \varphi_1' )'
  - \frac{N-1}{r}\, |\varphi_1'|^{p-2} \varphi_1'
  = \lambda_1\, m(r)\, \varphi_1^{p-1} \quad\mbox{for } r>0 ;
\\
  \mbox{subject to}\quad
  \int_0^\infty |\varphi_1'(r)|^p\, r^{N-1} \,{\rm d}r < \infty
    \quad\mbox{and }\quad
  \varphi_1(r) \to 0 \mbox{ as } r\to \infty .
\end{split}
\end{equation*}
%
It can be further rewritten as
%
\begin{equation}
\label{ev:phi_1.rad}
\begin{split}
{}- ( r^{N-1}\, |\varphi_1'|^{p-2} \varphi_1' )'
  = \lambda_1\, m(r)\, r^{N-1}\, \varphi_1^{p-1}
    \quad\mbox{for } r>0 ;
\\
  \varphi_1'(r)\to 0 \mbox{ as } r\to 0 \quad\mbox{and }\quad
  \varphi_1(r) \to 0 \mbox{ as } r\to \infty .
\end{split}
\end{equation}
%
Recalling hypothesis \eqref{hyp:m>0},
from \eqref{ev:phi_1.rad} we can deduce the following simple facts.


%%%%%    LEMMA - phi_1' < 0 and Simple Facts    %%%%

\begin{lemma}\label{lem-phi_1'}
Let $1<p<N$ and let hypothesis {H} be satisfied.
Then the function
%
\begin{math}
  r\mapsto r^{ \frac{N-1}{p-1} } \varphi_1'(r) \colon \mathbb{R}_+\to \mathbb{R}
\end{math}
%
is continuous and decreasing, and satisfies
$\varphi_1'(r) < 0$ for all $r>0$.
%
\end{lemma}

To determine the asymptotic behavior of
$\varphi_1(r)$ as $r\to \infty$,
we will investigate the corresponding nonlinear eigenvalue problem
\eqref{ev:phi_1.rad}
in Appendix~\ref{s:Asymptotic}.
Higher smoothness of $\varphi_1\colon \mathbb{R}_+\to (0,\infty)$
can be obtained directly by integrating equation \eqref{ev:phi_1.rad}:
$\varphi_1\in C^{1,\beta}(\mathbb{R}_+)$ with
$\beta = \min\{ 1 ,\, \frac{1}{p-1} \}$.
We refer to
{\sc Man\'a\-se\-vich} and {\sc Tak\'a\v{c}}
\cite[Eq.~(33)]{ManTakac} for details.

\subsection{Notation}
\label{ss:Notation}

The closure and boundary of a set $S\subset \mathbb{R}^N$
are denoted by $\overline{S}$ and $\partial S$, respectively.
We denote by
$B_{\varrho}\eqdef \{ x\in \mathbb{R}^N\colon |x| < \varrho\}$
the ball of radius $0 < \varrho < \infty$.

All Banach and Hilbert spaces used in this article are real.
Given an integer $k\geq 0$ and $0\leq \alpha\leq 1$,
we denote by
$C^{k,\alpha}(\mathbb{R}^N)$ the linear space of
all $k$-times continuously differentiable functions
$u\colon \mathbb{R}^N\to \mathbb{R}$ whose all (classical) partial derivatives
of order $\leq k$ are
locally $\alpha$-H\"older continuous on $\mathbb{R}^N$.
As usual, we abbreviate
$C^k(\mathbb{R}^N) \equiv C^{k,0}(\mathbb{R}^N)$.
The linear subspace of $C^k(\mathbb{R}^N)$
consisting of all $C^k$ functions
$u\colon \mathbb{R}^N\to \mathbb{R}$ with compact support is denoted by
$C_{\mathrm{c}}^k(\mathbb{R}^N)$.

For $1<p<2$ we denote by $\mathcal{D}_{\varphi_1}$
the normed linear space of all functions
$u\in D^{1,2}(\mathbb{R}^N)$ whose norm
%
\begin{equation}
 \| u\|_{ \mathcal{D}_{\varphi_1} } \eqdef
  \Big( \int_{\mathbb{R}^N}
    |\varphi_1'(|x|)|^{p-2} |\nabla u(x)|^2 \,{\rm d}x
  \Big)^{1/2}
\label{Q.norm}
\end{equation}
%
is finite.
Hence, the imbedding
$\mathcal{D}_{\varphi_1} \hookrightarrow D^{1,2}(\mathbb{R}^N)$
is continuous.
For $p=2$ we set
$\mathcal{D}_{\varphi_1} = D^{1,2}(\mathbb{R}^N)$.
Finally, for $2<p<N$ we define
$\mathcal{D}_{\varphi_1}$ to be the completion of
$D^{1,p}(\mathbb{R}^N)$ in the norm \eqref{Q.norm}.
Thus, the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$
is continuous.

For $1<p<N$ we denote by
$\mathcal{H}_{\varphi_1}$
the weighted Lebesgue space of all measurable functions
$u\colon \mathbb{R}^N\to \mathbb{R}$ with the norm
%
\[
 \| u\|_{ \mathcal{H}_{\varphi_1} } \eqdef
  \Big( \int_{\mathbb{R}^N} |u|^2\, \varphi_1^{p-2}\, m \,{\rm d}x
  \Big)^{1/2}
  < \infty
\]
%
and with the inner product
%
\[
 (u,v)_{ \mathcal{H}_{\varphi_1} } \eqdef
  \int_{\mathbb{R}^N} u\, v\, \varphi_1^{p-2}\, m \,{\rm d}x
    \quad\mbox{for } u,v\in \mathcal{H}_{\varphi_1} .
\]
%
The imbedding
$\mathcal{H}_{\varphi_1} \hookrightarrow L^{p}(\mathbb{R}^N; m)$
is continuous for $1 < p\leq 2$, and
$L^{p}(\mathbb{R}^N; m) \hookrightarrow \mathcal{H}_{\varphi_1}$
is continuous for $2\leq p < N$, by Lemma~\ref{lem-H^p.L^p}.
The Hilbert spaces
$\mathcal{D}_{\varphi_1}$ and $\mathcal{H}_{\varphi_1}$
will play an important role throughout this article.

We use the standard inner product in $L^2(\mathbb{R}^N)$ defined by
$\langle u,v \rangle\eqdef \int_{\mathbb{R}^N} uv \,{\rm d}x$
for $u,v\in L^2(\mathbb{R}^N)$.
This inner product induces a duality between the Lebesgue spaces
$L^{p}(\mathbb{R}^N; m)$ and $L^{p'}( \mathbb{R}^N; m^{-1/(p-1)} )$,
where $1\leq p < \infty$ and $1 < p'\leq \infty$ with
$\frac{1}{p} + \frac{1}{p'} = 1$,
and between the Sobolev space
$D^{1,p}(\mathbb{R}^N)$ and its dual $D^{-1,p'}(\mathbb{R}^N)$,
as well.
Similarly,
$\mathcal{D}_{\varphi_1}'$
($\mathcal{H}_{\varphi_1}'$, respectively)
stands for the dual space of
$\mathcal{D}_{\varphi_1}$
($\mathcal{H}_{\varphi_1}$).
We keep the same notation also for
the duality between the Cartesian products of such spaces.


\subsection{Linearization and quadratic forms}
\label{ss:Quadratic}

As usual,
$I$ is the identity matrix from $\mathbb{R}^{N\times N}$,
the tensor product ${\bf a}\otimes {\bf b}$ stands for
the $(N\times N)$-matrix
${\bf T} = (a_i b_j)_{i,j=1}^N$ whenever
${\bf a} = (a_i)_{i=1}^N$ and
${\bf b} = (b_i)_{i=1}^N$ are vectors from $\mathbb{R}^N$, and
$\langle \,\cdot\, ,\,\cdot\, \rangle_{\mathbb{R}^N}$
denotes the Euclidean inner product in $\mathbb{R}^N$.
We introduce the abbreviation
%
\begin{equation}
  {\bf A}({\bf a}) \eqdef |{\bf a}|^{p-2}
    \big(
    I + (p-2)\, \frac{ {\bf a}\otimes {\bf a} }{ |{\bf a}|^2 }
    \big)
  \quad\mbox{for } {\bf a}\in \mathbb{R}^N\setminus \{ \mathbf{0}\} .
\label{def.A=F''}
\end{equation}
%
We set
${\bf A}(\mathbf{0}) \eqdef \mathbf{0}\in \mathbb{R}^{N\times N}$
for all $1 < p < \infty$.
For ${\bf a}\not= \mathbf{0}$,
${\bf A}({\bf a})$ is a positive definite, symmetric matrix.
The spectrum of the matrix
$|{\bf a}|^{2-p} {\bf A}({\bf a})$
consists of the eigenvalues $1$ and $p-1$.
For all
${\bf a}, {\bf v}\in \mathbb{R}^N\setminus \{ \mathbf{0}\}$
we thus obtain
%
\begin{equation}
  0 <
  \min\{ 1,\,p-1\}
  \leq
  \frac{ \langle {\bf A}({\bf a}) {\bf v}, {\bf v} \rangle_{\mathbb{R}^N} }%
       { |{\bf a}|^{p-2} |{\bf v}|^2 }
  \leq
  \max\{ 1,\,p-1\} .
\label{A.ellipt}
\end{equation}
%

The following auxiliary inequalities are
Lemma~A.2 ($p\geq 2$) and Remark~A.3 ($p<2$) from
{\sc Tak\'a\v{c}} \cite[p.~235]{Takac-1}, respectively;
their proofs are straightforward.
First, for any $2\leq p < \infty$,
there exists a constant $c_p > 0$, such that for arbitrary vectors
${\bf a}, {\bf b}, {\bf v}\in \mathbb{R}^N$ we have
%
\begin{equation}
\label{1-s.A.geom:p>2}
\begin{split}
       c_p\cdot
    \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}|
    \big)^{p-2} |{\bf v}|^2
& \leq
    \int_0^1
    \langle {\bf A}({\bf a} + s {\bf b}) {\bf v}, {\bf v} \rangle
      (1-s) \,{\rm d}s \\
& \leq \frac{p-1}{2}
    \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}|
    \big)^{p-2} |{\bf v}|^2 .
\end{split}
\end{equation}
%
On the other hand, given any $1 < p < 2$,
there exists a constant $c_p > 0$, such that for arbitrary vectors
${\bf a}, {\bf b}, {\bf v}\in \mathbb{R}^N$, with
$|{\bf a}| + |{\bf b}| > 0$, we have
%
\begin{equation}
\label{1-s.A.geom:p<2}
\begin{split}
      \frac{p-1}{2}
    \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}|
    \big)^{p-2} |{\bf v}|^2
& \leq
    \int_0^1
    \langle {\bf A}({\bf a} + s {\bf b}) {\bf v}, {\bf v} \rangle
      (1-s) \,{\rm d}s
\\
& \leq c_p\cdot
    \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}|
    \big)^{p-2} |{\bf v}|^2 .
\end{split}
\end{equation}
%
These inequalities are needed to treat the linearization of
$-\Delta_p$ at $\varphi_1$ below.

Next, as in \cite[Sect.~1]{Takac-1},
we rewrite the first and second terms of the energy functional
$\mathcal{J}_{\lambda_1}$ using the integral forms of
the first- and second\--order Taylor formulas;
we set
%
\begin{equation}
 \mathcal{F}(u) \eqdef
    \frac{1}{p}
    \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
  - \frac{\lambda_1}{p}
    \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x ,\quad
    u\in D^{1,p}(\mathbb{R}^N) .
\label{def.J}
\end{equation}
%
We need to treat the Taylor formulas for $p\geq 2$ and $1<p<2$
separately.

\noindent{\it Case\/} $p\geq 2$.
Let $\phi\in D^{1,p}(\mathbb{R}^N)$ be arbitrary.
We take advantage of eq.~\eqref{e:varphi.l_1} to obtain
$\mathcal{J}(\varphi_1) = 0$ and consequently
%
\begin{equation} \label{J'.phi_1}
\begin{aligned}
   \mathcal{F}(\varphi_1 + \phi)
& = \int_0^1 \frac{{\rm d}}{{\rm d}s}\,
    \mathcal{F}(\varphi_1 + s\phi) \,{\rm d}s \\
& = \int_0^1
    \int_{\mathbb{R}^N} |\nabla (\varphi_1 + s\phi)|^{p-2}
                 \nabla (\varphi_1 + s\phi) \cdot \nabla\phi
    \,{\rm d}x \,{\rm d}s\\
&\quad   - \lambda_1 \int_0^1
    \int_{\mathbb{R}^N} |\varphi_1 + s\phi|^{p-2}
                 (\varphi_1 + s\phi) \phi\, m \,{\rm d}x \,{\rm d}s .
\end{aligned}
\end{equation}
%
Similarly, applying  \eqref{e:varphi.l_1} once again, i.e.,
$\mathcal{F}'(\varphi_1) = 0$, we get
%
\begin{equation}
  \mathcal{F}(\varphi_1 + \phi) = \mathcal{Q}_\phi (\phi,\phi)
\label{J''.phi_1}
\end{equation}
%
where $\mathcal{Q}_\phi$
is the symmetric bilinear form on the Cartesian product
$[ D^{1,p}(\mathbb{R}^N) ]^2$ defined as follows:
Given any fixed $\phi\in D^{1,p}(\mathbb{R}^N)$,
we set
%
\begin{equation} \label{def.Q}
\begin{aligned}
  \mathcal{Q}_\phi (v,w) &\eqdef
    \int_{\mathbb{R}^N}
    \Big\langle
  \big[
    \int_0^1 {\bf A}
  ( \nabla (\varphi_1 + s\phi)) (1-s) \,{\rm d}s
  \big] \nabla v,\, \nabla w
    \Big\rangle_{\mathbb{R}^N} \,{\rm d}x \\
&\quad
  - \lambda_1 (p-1)
    \int_{\mathbb{R}^N}
    \big[
    \int_0^1 |\varphi_1 + s\phi|^{p-2} (1-s) \,{\rm d}s
    \big] v\, w\, m \,{\rm d}x
\end{aligned}
\end{equation}
%
for all $v,w\in D^{1,p}(\mathbb{R}^N)$.
In particular, when $v\equiv w$ in $\mathbb{R}^N$,
one obtains the quadratic form $\mathcal{Q}_\phi (v,v)$.
If also $\phi\equiv 0$ then
%
\begin{equation}
 \mathcal{Q}_0 (v,v)
  = \frac12
    \int_{\mathbb{R}^N}
    \left\langle {\bf A} (\nabla\varphi_1) \nabla v,\, \nabla v
    \right\rangle_{\mathbb{R}^N} \,{\rm d}x
  - \frac12\lambda_1 (p-1)
    \int_{\mathbb{R}^N} v^2\, \varphi_1^{p-2}\, m \,{\rm d}x .
\label{def.Q_0}
\end{equation}
%
The imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$
being dense,
we extend the domain of the symmetric bilinear form
$\mathcal{Q}_0$ defined by~\eqref{def.Q_0} to all of
$\mathcal{D}_{\varphi_1}\times \mathcal{D}_{\varphi_1}$;
see e.g.\
{\sc Kato} \cite[Chapt.~VI, {\S}1.3, p.~313]{Kato}.

Note that, due to the radial symmetry of $\varphi_1$,
formula \eqref{def.A=F''} yields
%
\begin{equation}
  {\bf A} (\nabla\varphi_1) = | \varphi_1'(r) |^{p-2}
    \big(
    I + (p-2)\, \frac{ x\otimes x }{ r^2 }
    \big)
  \quad\mbox{with }\quad
    \nabla\varphi_1 = \varphi_1'(r)\, \frac{x}{r}
\label{phi_1.A=F''}
\end{equation}
%
for every $x\in \mathbb{R}^N$ with $r = |x| > 0$.
Furthermore,
our definition~\eqref{def.lam_1} of $\lambda_1$ and
eq.~\eqref{J''.phi_1} guarantee
$\mathcal{Q}_{t\phi}(\phi,\phi) \geq 0$
for all $t\in \mathbb{R}\setminus \{ 0\}$.
Letting $t\to 0$ we arrive at
%
\begin{equation}
  \mathcal{Q}_0 (\phi,\phi) \geq 0
    \quad\mbox{for all }\, \phi\in D^{1,p}(\mathbb{R}^N) .
\label{Q.geq.0}
\end{equation}
%

\noindent {\it Case\/} $1<p<2$.
Since $\mathcal{D}_{\varphi_1} \hookrightarrow D^{1,p}(\mathbb{R}^N)$
in this case, given any fixed $\phi\in D^{1,p}(\mathbb{R}^N)$,
we define the symmetric bilinear form $\mathcal{Q}_\phi$
on the Cartesian product
$\mathcal{D}_{\varphi_1} \times \mathcal{D}_{\varphi_1}$
by formula \eqref{def.Q}.
Notice that the first integral in \eqref{def.Q}
converges absolutely by inequality \eqref{1-s.A.geom:p<2}.
The absolute convergence of the second integral in \eqref{def.Q}
is obtained by similar arguments using also
the continuity of the imbedding
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$,
by Lemma~\ref{lem-Compact}.

\section{Main results}
\label{s:Main}

Recall that $1<p<N$ throughout this article unless indicated otherwise.

\subsection{Statements of Theorems}
\label{ss:Theorems}
The following two theorems are
the main results of our present article.

%%    EXISTENCE of a Solution (Theorem)

\begin{theorem}\label{thm-Exist:p>2}
Let $2\leq p < N$.
If $f\in \mathcal{D}_{\varphi_1}'$ satisfies
$\langle f, \varphi_1\rangle = 0$,
then problem~\eqref{e:BVP.l_1}
possesses a weak solution $u\in D^{1,p}(\mathbb{R}^N)$.
%
\end{theorem}


This is a part of the Fredholm alternative for
$-\Delta_p$ at $\lambda_1$.
The proof is given in Section~\ref{s:pr-Exist:p>2}.
In a bounded domain $\Omega\subset \mathbb{R}^N$,
this theorem is due to
{\sc Fleckinger} and {\sc Tak\'a\v{c}}
\cite[Theorem 3.3, p.~958]{FleckTakac-1}.

The orthogonality condition
$\langle f, \varphi_1\rangle = 0$
is sufficient, but {\em not\/} necessary
to obtain existence for problem~\eqref{e:BVP.l_1}
provided $p\neq 2$,
according to recent results obtained in
{\sc Dr\'abek}, {\sc Girg} and
{\sc Man\'a\-se\-vich} \cite[Theorem 1.3]{DrabGirgMan}
for $N=1$, in
{\sc Dr\'abek} and {\sc Holubov\'a} \cite[Theorem 1.1]{DrabHolub}
for any $N\geq 1$ and $1<p<2$, and in
{\sc Tak\'a\v{c}} \cite[Theorems 3.1 and 3.5]{Takac-2}
for any $N\geq 1$.


%%%%%    EXISTENCE of a Solution (Example)

\begin{example}\label{exam-Exist:p>2}
\begingroup\rm
For $2\leq p < N$, the hypothesis
$f\in \mathcal{D}_{\varphi_1}'$ is fulfilled, for example, if
$f = f_1 + f_2$ where $f_1\in L^2(B_\varepsilon; m^{-1})$ and
$f_1\equiv 0$ in $\mathbb{R}^N\setminus B_\varepsilon$, and
$f_2\equiv 0$ in $B_\varepsilon$ and
%
\begin{math}
  f_2\in  L^2\big( \mathbb{R}^N\setminus B_\varepsilon; r^{ - N + \frac{N-p}{p-1} } \big)
\end{math}
%
for some $0 < \varepsilon\leq 1$.
This claim follows from the imbeddings in Lemma~\ref{lem-Compact}
combined with the asymptotic formulas in
Proposition~\ref{prop-Asympt}, where
%
\begin{math}
  \mathcal{H}_{\varphi_1}' =
  L^{2}\big( \mathbb{R}^N; \varphi_1^{2-p} m^{-1} \big)
\end{math}
%
is the dual space of $\mathcal{H}_{\varphi_1}$, and
$L^{2}\left( \mathbb{R}^N; |\varphi_1'|^{-p} \varphi_1^2 \right)$
is the dual space of
$L^{2}\left( \mathbb{R}^N; |\varphi_1'|^p \varphi_1^{-2} \right)$.
\endgroup
\end{example}


%%%%%    Fredholm ALTERNATIVE at $\lambda_1$ (Theorem)

\begin{theorem}\label{thm-Exist:p<2}
Let $N\geq 2$ and\/ $1 < p < 2$.
Assume that
$f^{\#}\in D^{-1,p'}(\mathbb{R}^N)$ satisfies
$\langle f^{\#}, \varphi_1\rangle = 0$ and
$f^{\#}\not\equiv 0$ in $\mathbb{R}^N$.
Then there exist two numbers
$\delta\equiv \delta(f^{\#}) > 0$ and
$\varrho\equiv \varrho(f^{\#}) > 0$
such that problem \eqref{e:BVP.l} with
$f = f^{\#} + \zeta\, m\varphi_1^{p-1}$
has at least one solution whenever
$\lambda\in (\lambda_1 - \delta, \lambda_1 + \delta)$
and $\zeta\in (-\varrho,\varrho)$.
%
\end{theorem}

The proof of this theorem is given in Section~\ref{s:pr-Exist:p<2}.

%%%%%    Fredholm ALTERNATIVE at $\lambda_1$ (Remark)

\begin{remark}\label{rem-Exist:p<2}
\begingroup\rm
In the situation of {\rm Theorem \ref{thm-Exist:p<2}}, if
$\lambda\in (\lambda_1 - \delta, \lambda_1)$
and
$\zeta\in (-\varrho,\varrho)$,
then problem \eqref{e:BVP.l} has {\it at least three\/} solutions
$u_1, u_2, u_3\in D^{1,p}(\mathbb{R}^N)$, such that
%
\[
   \int_{\mathbb{R}^N} u_2\, \varphi_1^{p-1}\, m \,{\rm d}x
  < \int_{\mathbb{R}^N} u_1\, \varphi_1^{p-1}\, m \,{\rm d}x
  < \int_{\mathbb{R}^N} u_3\, \varphi_1^{p-1}\, m \,{\rm d}x ,
\]
%
$u_1$ is a saddle point
(which will be obtained in the proof of Theorem \ref{thm-Exist:p<2})
and $u_2, u_3$ are local minimizers for the functional
$\mathcal{J}_{\lambda}$ on $D^{1,p}(\mathbb{R}^N)$.
The proof of this claim is given in
Section \ref{s:pr-Exist:p<2}, {\S}\ref{ss:Fredh_Mount},
after the proof of Theorem \ref{thm-Exist:p<2}.
\endgroup
\end{remark}


%%%%%    Fredholm ALTERNATIVE at $\lambda_1$ (Example)

\begin{example}\label{exam-Exist:p<2}
\begingroup\rm
For $1 < p\leq 2$, the hypothesis
$f\in \mathcal{D}_{\varphi_1}'$ is fulfilled if
$|x|\, f(x)\in L^{p'}(\mathbb{R}^N)$ with $p'= p/(p-1)$,
by the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; |x|^{-p})$
in Lemma~\ref{lem-D^p.L^p}.
\endgroup
\end{example}


The proofs of both theorems above hinge on
the following imbeddings with weights.


%%%%%    PROPOSITION - Compact Imbedding

\begin{proposition}\label{prop-Compact}
Let $1<p<N$ and let hypothesis \eqref{hyp:m>0} be satisfied.
Then the following two imbeddings are compact:
%
\begin{itemize}
%
\item[{\rm (a)}]
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$;
%
\item[{\rm (b)}]
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$.
\end{itemize}
\end{proposition}

The proof of this proposition is given in Section~\ref{s:pf-Compact}.
The reader is referred to
{\sc Berger} and {\sc Schechter}
\cite[Proof of Theorem 2.4, p.~277]{BergSchecht},
{\sc Fleckinger}, {\sc Gossez}, and {\sc de~Th\'elin}
\cite[Lemma 2.3]{FGdT},
or {\sc Schechter} \cite{Schecht-1, Schecht-2}
for related imbeddings and compactness results.


\subsection{Properties of the corresponding energy functional}
\label{ss:J-Energy}

Weak solutions in $D^{1,p}(\mathbb{R}^N)$ to
the Dirichlet boundary value problem \eqref{e:BVP.l_1}
with $f\in D^{-1,p'}(\mathbb{R}^N)$
correspond to critical points of the energy functional
$\mathcal{J}_{\lambda_1} \colon D^{1,p}(\mathbb{R}^N)\to \mathbb{R}$
defined in \eqref{def.jl} with $\lambda = \lambda_1$.
Owing to the imbeddings in Proposition~\ref{prop-Compact},
all expressions in \eqref{def.jl} are meaningful.
For the cases $2\leq p < N$ and $1 < p < 2\leq N$,
the geometry of the functional $\mathcal{J}_{\lambda_1}$
is completely different; cf.\
{\sc Fleckinger} and {\sc Tak\'a\v{c}}
\cite[Theorem 3.1, p.~957]{FleckTakac-1}
and
{\sc Dr\'abek} and {\sc Holubov\'a}
\cite[Theorem 1.1, p.~184]{DrabHolub},
respectively, in a bounded domain $\Omega\subset \mathbb{R}^N$.

In the former case, we have the following analogue of
the {\em improved Poincar\'e inequality\/} from
\cite[Theorem 3.1, p.~957]{FleckTakac-1},
which is of independent interest.


%%%%%    An Improved Poincare INEQUALITY (Lemma)

\begin{lemma}\label{lem-Poincare}
Let $2\leq p < N$ and let hypothesis \eqref{hyp:m>0} be satisfied.
Then there exists a constant $c\equiv c(p,m) > 0$ such that
the inequality
%
\begin{equation}
\begin{split}
&   \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |u|^p\, m(x) \,{\rm d}x \\
& \geq
  c \Big(
    | u^\parallel |^{p-2}
    \int_{\mathbb{R}^N}
    |\nabla\varphi_1(x)|^{p-2} |\nabla u^\top|^2 \,{\rm d}x
  + \int_{\mathbb{R}^N} |\nabla u^\top|^p \,{\rm d}x
    \Big)
\end{split}
\label{e:Poincare}
\end{equation}
%
holds for all\/ $u\in D^{1,p}(\mathbb{R}^N)$.
%
\end{lemma}

Here, a function $u\in D^{1,p}(\mathbb{R}^N)$ is decomposed as
the direct sum \eqref{ortho:u}.
If the constant $c$ in~\eqref{e:Poincare} is replaced by zero,
one obtains
the classical Poincar\'e inequality; see e.g.\
{\sc Gilbarg} and {\sc Trudinger}
\cite[Ineq.\ (7.44), p.~164]{GilbargTrud}.
In analogy with the case $p=2$,
the {\em improved Poincar\'e inequality\/} \eqref{e:Poincare}
guarantees the solvability of
the Cauchy boundary value problem \eqref{e:BVP.l_1}
in the special case when
$f\in \mathcal{D}_{\varphi_1}'$ satisfies
$\langle f, \varphi_1\rangle = 0$.

On the other hand, the ``singular'' case $1 < p < 2\leq N$
is much different and has to be treated by
a minimax method introduced in
{\sc Tak\'a\v{c}} \cite[Sect.~7]{Takac-1}.
It uses the fact that the functional $\mathcal{J}_{\lambda_1}$
still remains coercive on
%
\begin{equation}
 D^{1,p}(\mathbb{R}^N)^\top \eqdef
    \Big\{
  u\in D^{1,p}(\mathbb{R}^N) \colon
    \int_{\mathbb{R}^N} u\, \varphi_1^{p-1}\, m \,{\rm d}x = 0
    \Big\} ,
\label{def.D^top}
\end{equation}
%
the complement of $\mathop{\rm lin} \{ \varphi_1\}$ in $D^{1,p}(\mathbb{R}^N)$
with respect to the direct sum~\eqref{ortho:u}, viz.\
%
\begin{math}
  D^{1,p}(\mathbb{R}^N) \penalty-1000 =
  \mathop{\rm lin} \{ \varphi_1\} \oplus D^{1,p}(\mathbb{R}^N)^\top
\end{math}.
%

The following notion introduced in
{\sc Dr\'abek} and {\sc Holubov\'a}
\cite[Def.\ 2.1, p.~185]{DrabHolub} is crucial.


%%%%%    Simple SADDLE Point Geometry (Definition)

\begin{definition}\label{def-Saddle_Geom}
\begingroup\rm
We say that a continuous functional
$\mathcal{E}\colon D^{1,p}(\mathbb{R}^N)\to \mathbb{R}$
has a {\em simple saddle point geometry\/}
if we can find $u,v\in D^{1,p}(\mathbb{R}^N)$ such that
%
\begin{gather*}
 \int_{\mathbb{R}^N} u\, \varphi_1^{p-1}\, m \,{\rm d}x < 0 <
  \int_{\mathbb{R}^N} v\, \varphi_1^{p-1}\, m \,{\rm d}x
    \quad\mbox{and }
\\
 \max\{ \mathcal{E}(u) ,\, \mathcal{E}(v) \} <
  \inf \left\{ \mathcal{E}(w)\colon w\in D^{1,p}(\mathbb{R}^N)^\top \right\} .
\end{gather*}
%
\endgroup
\end{definition}

Note that on any continuous path
$\theta\colon [-1,1]\to D^{1,p}(\mathbb{R}^N)$ with
$\theta(-1) = u$ and $\theta(1) = v$ there is a point
$w = \theta(t_0)\in D^{1,p}(\mathbb{R}^N)^\top$
for some $t_0\in [-1,1]$.
Hence,
$\max\{ \mathcal{E}(u) ,\, \mathcal{E}(v) \} < \mathcal{E}(w)$
shows that the function
$\mathcal{E}\circ \theta\colon [-1,1]\to \mathbb{R}$
attains its maximum at some $t'\in (-1,1)$.

The following result is essential;
in fact it replaces Lemma~\ref{lem-Poincare}.
For a bounded domain $\Omega\subset \mathbb{R}^N$, it was shown in
{\sc Dr\'abek} and {\sc Holubov\'a}
\cite[Lemma 2.1, p.~185]{DrabHolub}.

%
%%%%%    Simple SADDLE Point Geometry (Lemma)
%
\begin{lemma}\label{lem-Saddle_Geom}
Let\/ $1 < p < 2\leq N$.
Assume $f\in D^{-1,p'}(\mathbb{R}^N)$ with\/
$\langle f, \varphi_1\rangle = 0$ and\/
$f\not\equiv 0$ in~$\mathbb{R}^N$.
Then the functional\/ $\mathcal{J}_{\lambda_1}$
has a simple saddle point geometry.
Moreover, it is unbounded from below on $D^{1,p}(\mathbb{R}^N)$.
%
\end{lemma}
%

Its proof will be given in
Section \ref{s:pr-Exist:p<2}, {\S}\ref{ss:Saddle_Geom}.

For $1<p<2$ we will obtain a weak solution to
problem~\eqref{e:BVP.l} by showing that
the ``minimax'' (or rather ``maximin'') expression
%
\begin{equation}
  \beta_{\lambda}\eqdef
    \sup_{a < \tau < b}\
    \inf_{ u^\top\in D^{1,p}(\mathbb{R}^N)^\top }
  \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top )
\label{def.b_lam}
\end{equation}
%
provides a critical value $\beta_{\lambda}$
for the energy functional
$\mathcal{J}_{\lambda}$ defined in~\eqref{def.jl}.
Here $a,b$ ($-\infty < a < 0 < b < \infty$)
are provided by
the simple saddle point geometry of $\mathcal{J}_{\lambda_1}$
established in Lemma \ref{lem-Saddle_Geom} above.
Formula~\eqref{def.b_lam} is justified by
Lemma~\ref{lem-Ext-Cone}
({\S}\ref{ss:Ext-Cone})
whenever
$-\infty < \lambda < \Lambda_\infty$ and $f\in D^{-1,p'}(\mathbb{R}^N)$.
We will provide a simple sufficient condition for
the criticality of $\beta_{\lambda}$ in
Lemma~\ref{lem-Crit-Value}
({\S}\ref{ss:Minimax}).
This condition is verified in the setting of our
Theorem \ref{thm-Exist:p<2}
as a consequence of Lemma~\ref{lem-Saddle_Geom}.


%%%%%    SOME IMBEDDINGS WITH WEIGHTS

\section{Proof of Proposition~\ref{prop-Compact}}
\label{s:pf-Compact}

To prove this proposition, we need a few preliminary results.

%
\subsection{Some imbeddings with weights}
\label{ss:Imbedding}

We begin with the classical Hardy inequality
({\sc Kufner} \cite[Theorem 5.2, p.~28]{Kufner})
which reads
%
\begin{equation}
\label{e:Hardy}
    \int_{\mathbb{R}^N} \big(\frac{|u(x)| }{ |x| }\big)^{p} \,{\rm d}x
  \leq \big(\frac{p}{N-p}\big)^{p}
    \int_{\mathbb{R}^N} |\nabla u|^{p} \,{\rm d}x ,
    \quad u\in D^{1,p}(\mathbb{R}^N) .
\end{equation}
%
In particular, the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; |x|^{-p})$
is continuous.

Next, we show the continuity of some more imbeddings.


%%%%%    LEMMA - Compact Imbedding

\begin{lemma}\label{lem-D^p.L^p}
Let\/ $1<p<N$ and let hypothesis \eqref{hyp:m>0} be satisfied.
Then the following imbeddings are continuous:
%
\begin{gather}
D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; |x|^{-p})
                 \hookrightarrow L^{p}(\mathbb{R}^N; m) ;
\label{imb:D^p.L^p-m}\\
D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p^*}(\mathbb{R}^N)
                 \hookrightarrow L^{p}(\mathbb{R}^N; m) ,
\label{imb:D^p.L^p^*}
\end{gather}
%
where $p^* = Np / (N-p)$
denotes the critical Sobolev exponent.
%
\end{lemma}
%
%
\begin{proof} The imbedding
$L^{p}(\mathbb{R}^N; |x|^{-p}) \hookrightarrow L^{p}(\mathbb{R}^N; m)$
follows from inequality~\eqref{ineq:m>0}.

By a classical result
({\sc Gilbarg} and {\sc Trudinger}
 \cite[Theorem 7.10, p.~166]{GilbargTrud}),
the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p^*}(\mathbb{R}^N)$
is continuous.
Notice that
$(p/p^*) + (p/N) = 1$.
Finally, given an arbitrary function
$u\in C_{\mathrm{c}}^0(\mathbb{R}^N)$,
we combine the H\"older inequality with \eqref{ineq:m>0}
to estimate
%
\begin{align*}
   \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x
  &\leq
    \Big( \int_{\mathbb{R}^N} |u|^{p^*} \,{\rm d}x \Big)^{ p/p^* }
    \Big( \int_{\mathbb{R}^N} m^{N/p} \,{\rm d}x \Big)^{ p/N }
\\
& \leq C
    \Big( \int_{\mathbb{R}^N} |u|^{p^*} \,{\rm d}x \Big)^{ p/p^* }
    \Big( \int_{\mathbb{R}^N}
      (1 + |x|)^{ - N ( 1 + \frac{\delta}{p} ) } \,{\rm d}x
    \Big)^{ p/N } .
\end{align*}
%
The continuity of the imbedding
$L^{p^*}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$
follows because
$C_{\mathrm{c}}^0(\mathbb{R}^N)$ is dense in $L^{p^*}(\mathbb{R}^N)$.
\end{proof}

%%%%%    LEMMA - Compact Imbedding

\begin{lemma}\label{lem-H^p.L^p}
Let hypothesis \eqref{hyp:m>0} be satisfied.
Then we have the following imbeddings:
%
\begin{itemize}
%
\item[{\rm (i)}]

$\mathcal{H}_{\varphi_1} \hookrightarrow L^p(\mathbb{R}^N; m)$
if\/ $1<p<2$;
%
\item[{\rm (ii)}]
$\mathcal{H}_{\varphi_1} = L^2(\mathbb{R}^N; m)$
if\/ $p=2$;
%
\item[{\rm (iii)}]
$L^p(\mathbb{R}^N; m) \hookrightarrow \mathcal{H}_{\varphi_1}$
if\/ $2<p<N$.
%
\end{itemize}
\end{lemma}

\begin{proof}
We need to distinguish between the cases
$1<p<2$ and $2<p<N$.

\noindent {\it Case\/} $p<2$.
Let $u\in C_{\mathrm{c}}^0(\mathbb{R}^N)$ be arbitrary.
We apply H\"older's inequality again to estimate
%
\begin{equation*}
    \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x \leq
    \Big( \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
    \Big)^{ p/2 }
    \Big( \int_{\mathbb{R}^N} \varphi_1^{p}\, m \,{\rm d}x
    \Big)^{ (2-p)/2 }
  = \| u\|_{ \mathcal{H}_{\varphi_1} }^p ,
\end{equation*}
%
by
$\int_{\mathbb{R}^N} \varphi_1^p\, m \,{\rm d}x = 1$.
The space $C_{\mathrm{c}}^0(\mathbb{R}^N)$ being dense in
$\mathcal{H}_{\varphi_1}$, the imbedding in Part~{\rm (i)} follows.

{\it Case\/} $p>2$.
As above, for $u\in C_{\mathrm{c}}^0(\mathbb{R}^N)$ we estimate
%
\begin{equation*}
    \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x \leq
    \Big( \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x
    \Big)^{ 2/p }
    \Big( \int_{\mathbb{R}^N} \varphi_1^{p}\, m \,{\rm d}x
    \Big)^{ (p-2)/p }
  = \| u\|_{ L^{p}(\mathbb{R}^N; m) }^2 .
\end{equation*}
%
The lemma is proved.
\end{proof}

%%%%%    LEMMA - Compact Imbedding

\begin{lemma}\label{lem-D_1.D^p}
Let hypothesis \eqref{hyp:m>0} be satisfied.
The following imbeddings hold true:
%
\begin{itemize}
%
\item[{\rm (i)}]
$\mathcal{D}_{\varphi_1} \hookrightarrow D^{1,p}(\mathbb{R}^N)$
if\/ $1<p<2$;
%
\item[{\rm (ii)}]
$\mathcal{D}_{\varphi_1} = D^{1,2}(\mathbb{R}^N)$
if\/ $p=2$;
%
\item[{\rm (iii)}]

$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$
if\/ $2<p<N$.
%
\end{itemize}
%
\end{lemma}
%
%
\begin{proof}
Again, we distinguish between the cases
$1<p<2$ and $2<p<N$.

\noindent {\it Case\/} $p<2$.
Let $u\in C_{\mathrm{c}}^1(\mathbb{R}^N)$ be arbitrary.
H\"older's inequality yields
%
\begin{equation*}
\begin{aligned}
&   \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
  = \int_{\mathbb{R}^N} |\nabla u|^p\, |\varphi_1'(r)|^{ (p-2) p/2 }\,
                                |\varphi_1'(r)|^{-(p-2) p/2 }
    \,{\rm d}x \\
& \leq
    \Big( \int_{\mathbb{R}^N} |\nabla u|^2\, |\varphi_1'|^{p-2} \,{\rm d}x
    \Big)^{ p/2 }
    \Big( \int_{\mathbb{R}^N} |\varphi_1'|^{p} \,{\rm d}x
    \Big)^{ (2-p)/2 } \\
& = \lambda_1^{ (2-p)/2 }\, \| u\|_{ \mathcal{D}_{\varphi_1} }^p ,
\end{aligned}
\end{equation*}
%
by
$\int_{\mathbb{R}^N} |\varphi_1'|^p \,{\rm d}x = \lambda_1$.
The desired imbedding in Part~{\rm (i)}
now follows from the density of
$C_{\mathrm{c}}^1(\mathbb{R}^N)$ in $\mathcal{D}_{\varphi_1}$.

\noindent {\it Case\/} $p>2$.
Given $u\in C_{\mathrm{c}}^0(\mathbb{R}^N)$, we estimate
\begin{align*}
\int_{\mathbb{R}^N} |\nabla u|^2\, |\varphi_1'(r)|^{p-2} \,{\rm d}x
&\leq  \Big(
    \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
    \Big)^{ 2/p }
    \Big( \int_{\mathbb{R}^N} |\varphi_1'|^{p} \,{\rm d}x
    \Big)^{ (p-2)/p }\\
  &= \lambda_1^{ (p-2)/p }\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2 .
\end{align*}
%
This proves the lemma.
\end{proof}

%%%%%    LEMMA - Compact Imbedding

\begin{lemma}\label{lem-Compact}
Let\/ $1<p<N$ and let hypothesis \eqref{hyp:m>0} be satisfied.
Then both imbeddings
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
and
%
\begin{math}
  \mathcal{D}_{\varphi_1} \hookrightarrow
  L^{2}\left( \mathbb{R}^N; |\varphi_1'|^p \varphi_1^{-2} \right)
\end{math}
%
are continuous.
%
\end{lemma}
%
%
\begin{proof}
We need to distinguish between the cases
$1<p<2$ and $2\leq p < N$.

{\it Case\/} $p<2$.
Since
$\mathcal{D}_{\varphi_1} \hookrightarrow D^{1,2}(\mathbb{R}^N)$
with
%
\begin{math}
  \varphi_1'(r)\, r^{ \frac{N-1}{p-1} }
  \to {} - \frac{N-p}{p-1}\, c
\end{math}
%
as $r\to \infty$, by~\eqref{e:u'.infty},
the linear subspace of $\mathcal{D}_{\varphi_1}$
consisting of all functions with compact support is dense in
$\mathcal{D}_{\varphi_1}$.
So take an arbitrary function
$u\in \mathcal{D}_{\varphi_1}$ with compact support.
Using $u\in D^{1,2}(\mathbb{R}^N)$
and the properties of $\varphi_1$,
we deduce that both integrals below converge:
%
\[
    \int_{\mathbb{R}^N} u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2}
    \,{\rm d}x < \infty
  \quad\mbox{and }\quad
    \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
  < \infty .
\]
%
Consequently, we are allowed to apply the weak formulation of
the eigenvalue problem \eqref{e:varphi.l_1}
with the test function
$u^2 / \varphi_1 \in D^{1,1}(\mathbb{R}^N)$ to compute
%
\begin{align*}
&  \lambda_1\int_{\mathbb{R}^N}
    u(x)^2\, \varphi_1(r)^{p-2}\, m(r) \,{\rm d}x \\
&= \int_{\mathbb{R}^N}
    u(x)^2\, \varphi_1(r)^{-1} (- \Delta_p \varphi_1) \,{\rm d}x \\
& = \int_{\mathbb{R}^N}
    | \varphi_1'(r) |^{p-2}\, \varphi_1'(r)\,
    \frac{x}{r}\cdot \nabla \big(\frac{u^2}{\varphi_1}\big)
    \,{\rm d}x \\
& = 2 \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2}\, \varphi_1'\,
    \frac{\partial u}{\partial r}\, \frac{u}{\varphi_1}
    \,{\rm d}x
  - \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2}\, \varphi_1'\,
    \frac{\partial\varphi_1}{\partial r}\,
    \big(\frac{u}{\varphi_1}\big)^2 \,{\rm d}x .
\end{align*}
%
Adding the last integral and estimating the second last one by
the Cauchy\--Schwarz inequality, we arrive at
\begin{equation} \label{CS:Imbedd}
\begin{aligned}
&  \lambda_1\int_{\mathbb{R}^N}
    u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
  + \int_{\mathbb{R}^N}
    u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x \\
&\leq
  2 \Big( \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2}
    \big( \frac{\partial u}{\partial r}\big)^2 \,{\rm d}x
    \Big)^{1/2}
    \Big( \int_{\mathbb{R}^N}
    u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x
    \Big)^{1/2} \\
&\leq 2 \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2}
    \big(\frac{\partial u}{\partial r}\big)^2 \,{\rm d}x
  + \frac12 \int_{\mathbb{R}^N}
    u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x ,
\end{aligned}
\end{equation}
%
and therefore,
%
\begin{equation}
\label{ineq:Imbedd}
\begin{split}
&   \lambda_1\int_{\mathbb{R}^N}
    u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
  + \frac12 \int_{\mathbb{R}^N}
    u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x
\\
& \leq 2 \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2}\,
    \big(\frac{\partial u}{\partial r}\big)^2 \,{\rm d}x
  \leq 2\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2 .
\end{split}
\end{equation}
%
It follows that both imbeddings
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
and
%
\begin{math}
  \mathcal{D}_{\varphi_1} \hookrightarrow
  L^2\left( \mathbb{R}^N; |\varphi_1'|^p \varphi_1^{-2} \right)
\end{math}
%
are continuous.

\noindent {\it Case\/} $p\geq 2$.
The linear space $C_{\mathrm{c}}^1(\mathbb{R}^N)$ is dense in both
$D^{1,p}(\mathbb{R}^N)$ and $\mathcal{D}_{\varphi_1}$, by definition.
So take an arbitrary function
$u\in C_{\mathrm{c}}^1(\mathbb{R}^N)$.
The same procedure as for $p<2$ above leads us to
the inequalities in~\eqref{ineq:Imbedd}.
Again, both imbeddings
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
and
%
\begin{math}
  \mathcal{D}_{\varphi_1} \hookrightarrow
  L^2\left( \mathbb{R}^N; |\varphi_1'|^p \varphi_1^{-2} \right)
\end{math}
%
are continuous.
The lemma is proved.
\end{proof}

Next we will show that our hypothesis \eqref{hyp:m>0}
guarantees also the compactness of both imbeddings
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$
and
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
for $1<p<N$.
In order to prove this compactness,
given any $\varrho\in (0,\infty)$,
we introduce a cut\--off function
$\psi_{\varrho}\colon \mathbb{R}_+\to [0,1]$ as follows:
Take any $C^1$ function
$\psi_1\colon \mathbb{R}_+\to [0,1]$ such that
$\psi_1(r) = 1$ for $0\leq r\leq 1$,
$\psi_1(r) = 0$ for $2\leq r < \infty$, and
$\psi_1'(r)\leq 0$ for $1\leq r\leq 2$.
We define
%
\begin{math}
  \psi_{\varrho}(x)\equiv
  \psi_{\varrho}(r)\eqdef \psi_1(r / \varrho)
\end{math}
%
for all $x\in \mathbb{R}^N$ and $r = |x|$.
Notice that its radial derivative
$\psi_{\varrho}'(r) = (1 / \varrho)\, \psi_1'(r / \varrho)$
satisfies
%
\begin{equation}
  | \psi_{\varrho}'(r) | \leq C_1\, r^{-1}
  \quad\mbox{for all } r\geq 0 ,
\label{est:psi'}
\end{equation}
%
where
$C_1 = 2\cdot \sup_{\mathbb{R}_+} |\psi_1'| < \infty$
is a constant.
Obviously,
$\psi_{\varrho}(r) = 1$ for $0\leq r\leq \varrho$,
$\psi_{\varrho}(r) = 0$ for $2\varrho\leq r < \infty$, and
$\psi_{\varrho}'(r)\leq 0$ for $\varrho\leq r\leq 2\varrho$.
Now we define the corresponding cut\--off operator
%
\begin{math}
  T_{\varrho}\colon
  L_{\mathrm{loc}}^1(\mathbb{R}^N) \to L^1(\mathbb{R}^N)
\end{math}
%
by
$T_{\varrho} u\eqdef \psi_{\varrho} u$ for all
$u\in L_{\mathrm{loc}}^1(\mathbb{R}^N)$.
These linear operators are uniformly bounded
from $D^{1,p}(\mathbb{R}^N)$
($\mathcal{D}_{\varphi_1}$, respectively)
into itself for all $\varrho > 0$ sufficiently large
as is shown in the following lemma.

%%%%%    LEMMA - Cut-off: Compact Imbedding

\begin{lemma}\label{lem-Cut-off}
Let\/ $1<p<N$ and let hypothesis \eqref{hyp:m>0} be satisfied.
Then there exist constants $C_2 > 0$, $C_3 > 0$ and $R_1 > 0$,
such that for all\/ $\varrho\geq R_1$ we have
%
\begin{gather}
  \| \psi_{\varrho} u \|_{ D^{1,p}(\mathbb{R}^N) }
 \leq C_2\, \| u\|_{ D^{1,p}(\mathbb{R}^N) }
  \quad\mbox{for all } u\in D^{1,p}(\mathbb{R}^N) ;
\label{est:T_psi,p}\\
  \| \psi_{\varrho} u \|_{ \mathcal{D}_{\varphi_1} }
 \leq C_3\, \| u\|_{ \mathcal{D}_{\varphi_1} }
  \quad\mbox{for all } u\in \mathcal{D}_{\varphi_1} .
\label{est:T_psi}
\end{gather}
\end{lemma}
%

\begin{proof}
We give the proof for the case $1<p<2$ only and leave minor changes
for $2\leq p < N$ to the reader.
Let $\varrho > 0$.
For an arbitrary function $u\in D^{1,p}(\mathbb{R}^N)$ we have
$$
    \nabla (\psi_{\varrho} u)
  = \psi_{\varrho}(r)\, \nabla u(x)
  + u(x)\, \psi_{\varrho}'(r)\, r^{-1} x
  \quad\mbox{for $x\in \mathbb{R}^N$ and $r = |x|$. }
$$
Therefore, by the Minkowski inequality followed by
\eqref{est:psi'} and the Hardy inequality \eqref{e:Hardy},
we have
%
\begin{equation}
\label{ineq:T_psi,p}
\begin{aligned}
 \| \psi_{\varrho} u \|_{ D^{1,p}(\mathbb{R}^N) }
 &= \Big(
    \int_{\mathbb{R}^N} |\nabla (\psi_{\varrho} u)|^p \,{\rm d}x
    \Big)^{1/p}\\
& \leq
    \Big(
    \int_{\mathbb{R}^N} |\psi_{\varrho}|^p |\nabla u|^p \,{\rm d}x
    \Big)^{1/p}
  + \Big(
    \int_{\mathbb{R}^N} |\psi_{\varrho}'|^p |u|^p \,{\rm d}x
    \Big)^{1/p} \\
& \leq
    \| u\|_{ D^{1,p}(\mathbb{R}^N) }
  + C_1
    \Big(
    \int_{\mathbb{R}^N} |u(x)|^p\, |x|^{-p} \,{\rm d}x
    \Big)^{1/p}\\
&\leq C_2\, \| u\|_{ D^{1,p}(\mathbb{R}^N) } ,
\end{aligned}
\end{equation}
%
where $C_2 = 1 + p C_1 / (N-p)$.
This proves \eqref{est:T_psi,p}.

Similarly, for every $u\in \mathcal{D}_{\varphi_1}$ we have
%
\begin{equation}
\label{ineq:T_psi}
\begin{aligned}
\| \psi_{\varrho} u \|_{ \mathcal{D}_{\varphi_1} }
&= \big( \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2} |\nabla (\psi_{\varrho} u)|^2 \,{\rm d}x
    \big)^{1/2} \\
& \leq
    \Big(
    \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2} |\psi_{\varrho}|^2 |\nabla u|^2 \,{\rm d}x
    \Big)^{1/2}
  + \Big(
    \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2} |\psi_{\varrho}'|^2 u^2 \,{\rm d}x
    \Big)^{1/2}\\
& \leq
    \| u\|_{ \mathcal{D}_{\varphi_1} }
  + \Big(
    \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2} |\psi_{\varrho}'|^2 u^2 \,{\rm d}x
    \Big)^{1/2} .
\end{aligned}
\end{equation}
%
The last integral is estimated as follows.
Using the limit formula \eqref{e:u'/u.infty} we have
%
\begin{equation}
\label{e:psi'/psi.infty}
  \varphi_1^{-1} |\varphi_1'| \geq \frac{N-p}{ 2(p-1)r }
  \quad\mbox{for all } r\geq R_1 ,
\end{equation}
%
where $R_1 > 0$ is a sufficiently large constant.
We combine this inequality with \eqref{est:psi'}
to conclude that
%
\begin{equation}
\label{est:psi'.phi_1}
  | \psi_{\varrho}'(r) | \leq C_4\, \varphi_1^{-1} |\varphi_1'|
  \quad\mbox{for all } r\geq R_1 ,
\end{equation}
%
where
$C_4 = 2 (p-1) C_1 / (N-p)$.
Applying this estimate to the last integral in \eqref{ineq:T_psi},
and recalling
$\psi_{\varrho}'(r) = 0$ whenever $0\leq r\leq \varrho$,
for every $\varrho\geq R_1$ we get
%
\begin{equation*}
  \| \psi_{\varrho} u \|_{ \mathcal{D}_{\varphi_1} }
  \leq
    \| u\|_{ \mathcal{D}_{\varphi_1} }
  + C_4
    \Big( \int_{\mathbb{R}^N}
    |\varphi_1'|^p |\varphi_1|^{-2} u^2 \,{\rm d}x
    \Big)^{1/2} .
\end{equation*}
%
Finally, we invoke inequality \eqref{ineq:Imbedd}
to estimate the last integral.
The desired estimate \eqref{est:T_psi}
follows with the constant $C_3 > 0$ given by
$C_3 = 1 + 2 C_4$.
\end{proof}

Denoting by $J$ ($J_{\varphi_1}$, respectively)
the continuous imbedding \\
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$
($\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$),
we now show that the operators
%
\begin{equation*}
  J T_{\varrho}\colon D^{1,p}(\mathbb{R}^N) \to L^{p}(\mathbb{R}^N; m)
  \qquad
(
  J_{\varphi_1} T_{\varrho}\colon
  \mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1}
)
\end{equation*}
%
converge to $J$ ($J_{\varphi_1}$)
in the uniform operator topology as $\varrho\to \infty$.

%%%%%    LEMMA - Cut-off: Convergence

\begin{lemma}\label{lem-Cut-off:Conv}
Let\/ $1<p<N$ and let hypothesis \eqref{hyp:m>0} be satisfied.
Then, as $\varrho\to \infty$, we have
%
\begin{gather}
  \| (1 - \psi_{\varrho}) u \|_{ L^{p}(\mathbb{R}^N; m) } \to 0
 \quad\mbox{uniformly for }\
    \| u\|_{ D^{1,p}(\mathbb{R}^N) } \leq 1 ;
\label{est:I-T_psi,p} \\
  \| (1 - \psi_{\varrho}) u \|_{ \mathcal{H}_{\varphi_1} } \to 0
 \quad\mbox{uniformly for }\
    \| u\|_{ \mathcal{D}_{\varphi_1} } \leq 1 .
\label{est:I-T_psi}
\end{gather}
%
\end{lemma}
%
%
\begin{proof}
 From hypothesis \eqref{hyp:m>0} we get
%
\begin{equation*}
    m(r)\, r^p
  \leq \frac{C\, r^p}{ (1+r)^{p+\delta} }
  < \frac{C}{ (1+r)^{\delta} }
    \quad\mbox{for all } r>0 .
\end{equation*}
%
Hence, for any $\varrho > 0$,
%
\begin{align*}
    \int_{ |x|\geq \varrho } |u|^p\, m \,{\rm d}x
& \leq \frac{C}{ (1 + \varrho)^{\delta} }
    \int_{ |x|\geq \varrho } |u|^p\, |x|^{-p} \,{\rm d}x
\\
& \leq \frac{C}{ (1 + \varrho)^{\delta} }
    \big( \frac{p}{N-p}\big)^{p}
       \| u\|_{ D^{1,p}(\mathbb{R}^N) }^{p} ,
\end{align*}
%
by the Hardy inequality \eqref{e:Hardy}.
Letting $\varrho\to \infty$ we obtain
the convergence \eqref{est:I-T_psi,p}.

Similarly as above, we combine
hypothesis \eqref{hyp:m>0} and inequality \eqref{e:psi'/psi.infty}
to compare the weights
%
\begin{equation*}
    \frac{ \varphi_1(r)^{p-2}\, m(r) }
         { |\varphi_1'(r)|^{p}\, \varphi_1(r)^{-2} }
  \leq \frac{C_5\, r^p}{ (1+r)^{p+\delta} }
  < \frac{C_5}{ (1+r)^{\delta} }
    \quad\mbox{for all } r\geq R_1 ,
\end{equation*}
%
where
$$
  C_5 =\big( \frac{ 2 (p-1)}{N-p}\big)^p C .
$$
We use this inequality to estimate the second integral on
the left\--hand side in \eqref{ineq:Imbedd},
thus arriving at
%
\begin{equation*}
    \lambda_1\int_{\mathbb{R}^N}
    u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
  + \frac{ (1 + \varrho)^{\delta} }{ 2 C_5 }
    \int_{ |x|\geq \varrho }
    u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
  \leq 2\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2
\end{equation*}
%
for every $\varrho\geq R_1$.
Letting $\varrho\to \infty$ we obtain
the conclusion \eqref{est:I-T_psi} immediately.
\end{proof}


%
\subsection{Rest of the proof of Proposition~\ref{prop-Compact}}
\label{ss:pf-Compact}

According to Lemmas \ref{lem-D^p.L^p} and~\ref{lem-Compact},
it remains to show that the imbeddings
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$
and
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
are compact.
We take advantage of the well\--known approximation theorem
(see {\sc Kato} \cite[Chapt.~III, {\S}4.2, p.~158]{Kato})
which states that the set of all compact linear operators
$S\colon X\to Y$, where $X$ and $Y$ are Banach spaces,
is a Banach space.
In our setting this means that, by Lemma \ref{lem-Cut-off:Conv},
it suffices to show that the operators
%
\begin{equation*}
  J T_{\varrho}\colon D^{1,p}(\mathbb{R}^N) \to L^{p}(\mathbb{R}^N; m)
  \quad\mbox{and }\quad
  J_{\varphi_1} T_{\varrho}\colon
  \mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1} ,
\end{equation*}
%
respectively, are compact for each $\varrho > 0$ large enough.

Recall
$B_r = \{ x\in \mathbb{R}^N\colon |x|<r\}$ for $0<r<\infty$.
A function $u\in L^2(B_r)$ or
$u\in L^2( \mathbb{R}^N\setminus B_r ; m )$, respectively,
is naturally extended to all of $\mathbb{R}^N$ by setting $u(x) = 0$
for all $x\in \mathbb{R}^N\setminus B_r$ or for all $x\in B_r$.
We observe that
$$
  W_0^{1,p}(B_r) =
  \{ u\in D^{1,p}(\mathbb{R}^N) \colon
     u=0 \mbox{ almost everywhere in } \mathbb{R}^N\setminus B_r
  \}
$$
and set
$$
  \mathcal{D}_{\varphi_1}(B_r) \eqdef
  \{ u\in \mathcal{D}_{\varphi_1} \colon
     u=0 \mbox{ almost everywhere in } \mathbb{R}^N\setminus B_r
  \} .
$$
Clearly, $W_0^{1,p}(B_r)$
is a closed linear subspace of $D^{1,p}(\mathbb{R}^N)$
and the same is true of
$\mathcal{D}_{\varphi_1}(B_r)$ in $\mathcal{D}_{\varphi_1}$.

\noindent {\it Proof of\/} Part~{\rm (a)}.
By Lemma \ref{lem-Cut-off}, the cut\--off operators
%
\begin{equation*}
  T_{\varrho}\colon D^{1,p}(\mathbb{R}^N) \to W_0^{1,p}(B_{2\varrho})
  \subset D^{1,p}(\mathbb{R}^N)
\end{equation*}
%
are uniformly bounded for all $\varrho\geq R_1$.
Furthermore, the imbedding
$W_0^{1,p}(B_{2\varrho}) \hookrightarrow L^p(B_{2\varrho})$
being compact by Rellich's theorem, and
$L^p(B_{2\varrho}) \hookrightarrow L^{p}(\mathbb{R}^N; m)$
being continuous by \eqref{ineq:m>0}, we conclude that
%
\begin{math}
  J T_{\varrho}\colon D^{1,p}(\mathbb{R}^N) \to L^{p}(\mathbb{R}^N; m)
\end{math}
%
is compact as well, whenever $\varrho\geq R_1$.

\noindent {\it Proof of\/} Part~{\rm (b)}.
We need to treat the two cases
$1<p<2$ and $2\leq p < N$ separately.

\noindent {\it Case\/} $p<2$.
By Lemma \ref{lem-Cut-off}, the operators
%
\begin{math}
  T_{\varrho}\colon
  \mathcal{D}_{\varphi_1} \to \mathcal{D}_{\varphi_1}(B_{2\varrho})
  \subset \mathcal{D}_{\varphi_1}
\end{math}
%
are uniformly bounded for all $\varrho\geq R_1$.
Furthermore, the imbedding
%
\begin{math}
  \mathcal{D}_{\varphi_1}(B_{2\varrho}) \hookrightarrow
  W_0^{1,2}(B_{2\varrho})
\end{math}
%
is continuous by
$\gamma_1\eqdef \inf_{ (0,\infty) } |\varphi_1'|^{p-2} > 0$.
Finally, the imbedding
$W_0^{1,2}(B_{2\varrho}) \hookrightarrow L^2(B_{2\varrho})$
being compact by Rellich's theorem, and
$L^2(B_{2\varrho}) \hookrightarrow \mathcal{H}_{\varphi_1}$
being continuous by \eqref{ineq:m>0}, we conclude that
%
\begin{math}
  J_{\varphi_1} T_{\varrho}\colon
  \mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1}
\end{math}
%
is compact as well, whenever $\varrho\geq R_1$.

\noindent {\it Case\/} $p\geq 2$.
First, taking an arbitrary function
$u\in C^1(\mathbb{R}^N)$ with compact support, we derive
inequalities \eqref{CS:Imbedd} and \eqref{ineq:Imbedd}.
In particular, inequalities in \eqref{CS:Imbedd} entail
%
\begin{equation}\label{est:Imbedd:p>2}
\begin{aligned}
  \lambda_1\int_{\mathbb{R}^N}
    u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
&\leq
  2 \Big( \int_{\mathbb{R}^N}
    |\varphi_1'|^{p-2}
    \big(\frac{\partial u}{\partial r}\big)^2 \,{\rm d}x
    \Big)^{1/2}
    \Big( \int_{\mathbb{R}^N}
    u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x
    \Big)^{1/2}\\
&\leq 2\, \| u\|_{ \mathcal{D}_{\varphi_1} }
    \Big( \int_{\mathbb{R}^N}
    u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x
    \Big)^{1/2} .
\end{aligned}
\end{equation}
%
We need to show that, besides inequalities \eqref{ineq:Imbedd},
we have also
%
\begin{equation}
  \int_{B_R} |\varphi_1'|^{p}\, \varphi_1^{-2}\, u^2 \,{\rm d}x
  \leq 9\cdot \log\genfrac{(}{)}{}0{ \varphi_1(0) }{ \varphi_1(R) }
        \cdot \| u\|_{ \mathcal{D}_{\varphi_1} }^2
  \quad\mbox{for every } R>0 .
\label{R:Imbedd:p>2}
\end{equation}
%

To this end, fix any $x'\in \mathbb{R}^N$ with $|x'| = 1$,
and take $x = rx'$ with $0\leq r\leq R$.
We use eq.~\eqref{ev:phi_1.rad} to compute
%
\begin{align*}
&     r^{N-1}\, |\varphi_1'(r)|^{p-1}\, \varphi_1(r)^{-1}\, u(rx')^2
  = - \left(
      r^{N-1}\, |\varphi_1'|^{p-2} \varphi_1'
      \right) \varphi_1^{-1}\, u^2\\
& = - \int_0^r \frac{\partial}{\partial s}
      \left[
      s^{N-1}\, |\varphi_1'(s)|^{p-2} \varphi_1(s)'\,
      \varphi_1(s)^{-1}\, u(sx')^2
      \right]
      \,{\rm d}s\\
& = \lambda_1 \int_0^r
      m(s)\, s^{N-1}\, \varphi_1(s)^{p-2}\, u(sx')^2 \,{\rm d}s \\
& \quad   + \int_0^r
      s^{N-1}\, |\varphi_1'(s)|^{p}\, \varphi_1(s)^{-2}\,
      u(sx')^2 \,{\rm d}s \\
& \quad   + 2 \int_0^r
      s^{N-1}\, |\varphi_1'(s)|^{p-1}\, \varphi_1(s)^{-1}\,
      u(sx')\, \frac{\partial u}{\partial s}(sx') \,{\rm d}s .
\end{align*}
%
Estimating the last integral by
the Cauchy\--Schwarz inequality, we have
%
\begin{align*}
    r^{N-1}\, |\varphi_1'(r)|^{p-1}\, \varphi_1(r)^{-1}\, u(rx')^2
& \leq \lambda_1 \int_0^r
      m(s)\, \varphi_1(s)^{p-2}\, u(sx')^2\, s^{N-1} \,{\rm d}s
\\
&\quad   + 2 \int_0^r
      |\varphi_1'(s)|^{p}\, \varphi_1(s)^{-2}\, u(sx')^2\,
      s^{N-1} \,{\rm d}s
\\
&\quad   + \int_0^r |\varphi_1'(s)|^{p-2}\,
      \left( \frac{\partial u}{\partial s}(sx') \right)^2\,
      s^{N-1} \,{\rm d}s .
\end{align*}
%
Next, setting $y = sx'$,
we integrate this inequality with respect to $x'$ over the unit sphere
$S_1 = \partial B_1\subset \mathbb{R}^N$ endowed with the surface measure
$\sigma$ to get
%
\begin{equation}
\label{p-1:phi_1.rad}
\begin{aligned}
&     r^{N-1}\, |\varphi_1'(r)|^{p-1}\, \varphi_1(r)^{-1}
    \int_{S_1} u(rx')^2 \,{\rm d}\sigma(x')
\\
& \leq \lambda_1 \int_{B_r}
    u^2\, \varphi_1^{p-2}\, m \,{\rm d}y
  + 2 \int_{B_r}
    u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}y
\\
& \quad
  + \int_{B_r} |\varphi_1'|^{p-2}\,
    \big(\frac{\partial u}{\partial s}\big)^2 \,{\rm d}y
  \leq 8\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2
     +     \| u\|_{ \mathcal{D}_{\varphi_1} }^2
     = 9\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2 ,
\end{aligned}
\end{equation}
%
by ineq.~\eqref{ineq:Imbedd}.
Finally, upon multiplication by $-\varphi_1'/ \varphi_1$
followed by integration over $0\leq r\leq R$,
we arrive at the desired inequality \eqref{R:Imbedd:p>2}.

Again, by Lemma \ref{lem-Cut-off}, the operators
%
\begin{math}
  T_{\varrho}\colon
  \mathcal{D}_{\varphi_1} \to \mathcal{D}_{\varphi_1}(B_{2\varrho})
  \subset \mathcal{D}_{\varphi_1}
\end{math}
%
are uniformly bounded for all $\varrho\geq R_1$.
In order to show that
%
\begin{math}
  J T_{\varrho}\colon
  \mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1}
\end{math}
%
is compact, it suffices to verify that the imbedding
%
\begin{math}
  \mathcal{D}_{\varphi_1}(B_{2\varrho}) \hookrightarrow
  \mathcal{H}_{\varphi_1}
\end{math}
%
is compact.
So let $\varrho\geq R_1$ be fixed.

Consider an arbitrary bounded sequence
$\{ u_n\}_{n=1}^\infty$ in the Hilbert space
$\mathcal{D}_{\varphi_1}(B_{2\varrho})$.
Hence, there exists a weakly convergent subsequence
denoted again by $\{ u_n\}_{n=1}^\infty$, i.e.,
$u_n\rightharpoonup u$ in
$\mathcal{D}_{\varphi_1}(B_{2\varrho})$ as $n\to \infty$.
Replacing $u_n - u$ by $u_n$, we may assume
$u_n\rightharpoonup 0$ weakly in
$\mathcal{D}_{\varphi_1}(B_{2\varrho})$.
In addition, we may assume
$\| u_n\|_{ \mathcal{D}_{\varphi_1} } \leq 1$ for all $n=1,2,\dots$.
Next, we show that $u_n\to 0$ strongly in
$L^{2}\left( B_{2\varrho} ; |\varphi_1'|^p \varphi_1^{-2} \right)$.
Choose $\varepsilon > 0$.
Fix $R_0 > 0$ small enough, such that
\[
  9\cdot \log\big(\frac{\varphi_1(0)}{\varphi_1(R_0)}\big)
  \leq \frac{\varepsilon}{2} ,
\]
by
$\lim_{r\to 0} \varphi_1(r) = \varphi_1(0) > 0$.
Hence, inequality \eqref{R:Imbedd:p>2} entails
%
\begin{equation}
  \int_{B_{R_0}}
    |\varphi_1'|^{p}\, \varphi_1^{-2}\, u_n^2 \,{\rm d}x
  \leq \frac{\varepsilon}{2}
  \quad\mbox{for } n=1,2,\dots .
\label{R_0:Imbedd:p>2}
\end{equation}
%

Since
$\gamma_2\eqdef \inf_{ [ R_0, 2\varrho ] } |\varphi_1'|^{p-2} > 0$,
by Lemma~\ref{lem-phi_1'},
the sequence
$\{ u_n\}_{n=1}^\infty$ is bounded in the Sobolev space
$W^{1,2}( B_{2\varrho} \setminus B_{R_0} )$,
by inequalities \eqref{ineq:Imbedd}.
The imbedding
%
\begin{math}
  W^{1,2}( B_{2\varrho} \setminus B_{R_0} ) \hookrightarrow
  L^2    ( B_{2\varrho} \setminus B_{R_0} )
\end{math}
%
being compact by Rellich's theorem, we conclude that
$u_n\to 0$ strongly in
$L^2( B_{2\varrho} \setminus B_{R_0} )$.
Consequently, there is an integer $n_0\geq 1$ large enough,
such that
%
\begin{equation}
  \int_{ B_{2\varrho} \setminus B_{R_0} }
    |\varphi_1'|^{p}\, \varphi_1^{-2}\, u_n^2 \,{\rm d}x
  \leq \frac{\varepsilon}{2}
  \quad\mbox{for every } n\geq n_0 .
\label{r>R_0:Imbedd:p>2}
\end{equation}
%
We combine estimates
\eqref{R_0:Imbedd:p>2} and \eqref{r>R_0:Imbedd:p>2}
to obtain
%
\begin{equation*}
  \int_{ B_{2\varrho} }
    |\varphi_1'|^{p}\, \varphi_1^{-2}\, u_n^2 \,{\rm d}x
  \leq \varepsilon
  \quad\mbox{for every } n\geq n_0 .
\end{equation*}
%
This means that $u_n\to 0$ strongly in
$L^{2}\left( B_{2\varrho} ; |\varphi_1'|^p \varphi_1^{-2} \right)$.
Finally, from inequality \eqref{est:Imbedd:p>2}
we deduce $u_n\to 0$ strongly also in $\mathcal{H}_{\varphi_1}$.
Hence, the imbedding
%
\begin{math}
  \mathcal{D}_{\varphi_1}(B_{2\varrho}) \hookrightarrow
  \mathcal{H}_{\varphi_1}
\end{math}
%
is compact as claimed.

We have completed the proof of Proposition~\ref{prop-Compact}.


\section{Properties of the quadratization at $\varphi_1$}
\label{s:Auxiliary}

In this section we state a few analog results to those in
{\sc Tak\'a\v{c}} \cite[Sect.~4]{Takac-1}
that are employed later in the proofs of
Theorem~\ref{thm-Exist:p>2} and Lemma~\ref{lem-Poincare}.

Note that inequality~\eqref{A.ellipt} entails
%
\begin{equation}
\label{norm:A.equiv}
  \min\{ 1,\,p-1\}
    \| v\|_{ \mathcal{D}_{\varphi_1} }^2
  \leq \int_{\mathbb{R}^N}
    \langle {\bf A} (\nabla\varphi_1) \nabla v , \nabla v
    \rangle_{\mathbb{R}^N}
    \,{\rm d}x
  \leq
    \max\{ 1,\,p-1\}
    \| v\|_{ \mathcal{D}_{\varphi_1} }^2
\end{equation}
%
for $v\in \mathcal{D}_{\varphi_1}$.
Several important properties of $\mathcal{D}_{\varphi_1}$
are established below.
The following result is obvious.

%%%%%    Q - nonnegative (Lemma)

\begin{lemma}\label{lem-Q-posit}
We have
$\mathcal{Q}_0 (\varphi_1,\varphi_1) = 0$ and\/
$0\leq \mathcal{Q}_0 (v,v) < \infty$ for all\/
$v\in \mathcal{D}_{\varphi_1}$.
%
\end{lemma}
%

We denote by $\mathcal{A}_{\varphi_1}$
the Lax\--Milgram representation of the symmetric bilinear form
$2\cdot \mathcal{Q}_0$ on
$\mathcal{D}_{\varphi_1}\times \mathcal{D}_{\varphi_1}$
(see \cite[Chapt.~VI, Eq.\ (2.3), p.~323]{Kato}).
In our setting this means that
%
\begin{math}
  \mathcal{A}_{\varphi_1} \colon
  \mathcal{D}_{\varphi_1} \to \mathcal{D}_{\varphi_1}'
\end{math}
%
is a bounded linear operator such that
%
\begin{equation}
    \langle \mathcal{A}_{\varphi_1} v, w\rangle
  = 2\cdot \mathcal{Q}_0 (v,w)
  \quad\mbox{for all } v,w\in \mathcal{D}_{\varphi_1} .
\label{def.A}
\end{equation}
%
Identifying the dual space of
$\mathcal{D}_{\varphi_1}'$ with $\mathcal{D}_{\varphi_1}$
(see {\sc Yosida} \cite[Theorem IV.8.2, p.~113]{Yosida}),
we find that $\mathcal{A}_{\varphi_1}$
is selfadjoint in the following sense:
$$
    \langle \mathcal{A}_{\varphi_1} v, w\rangle
  = \langle v, \mathcal{A}_{\varphi_1} w\rangle
  \quad\mbox{for all } v,w\in \mathcal{D}_{\varphi_1} .
$$
Note that our definition of $\mathcal{Q}_0$ yields
$\mathcal{A}_{\varphi_1} \varphi_1 = 0$.
Since the imbedding
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
is compact,
the null space of $\mathcal{A}_{\varphi_1}$ denoted by
\[
  \ker(\mathcal{A}_{\varphi_1}) =
  \{ v\in \mathop{\rm dom}(\mathcal{A}_{\varphi_1})\colon
     \mathcal{A}_{\varphi_1} v = 0 \}
\]
is finite\--dimensional, by the Riesz\--Schauder theorem
\cite[Theorem III.6.29, p.~187]{Kato}.

Lemma~\ref{lem-Q-posit} provides
another variational formula for $\lambda_1$, namely,
%
\begin{equation}
  \lambda_1 = \inf
    \Big\{
  \frac{ \int_{\mathbb{R}^N}
    \langle {\bf A}(\nabla\varphi_1) \nabla u, \nabla u
    \rangle_{\mathbb{R}^N} \,{\rm d}x }%
       { (p-1) \int_{\mathbb{R}^N} |u|^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
  \colon 0\not\equiv u\in \mathcal{D}_{\varphi_1}
    \Big\} ,
\label{eq.lam_1}
\end{equation}
%
cf.\ eq.~\eqref{def.lam_1}.
This is a generalized Rayleigh quotient formula for
the first (smallest) eigenvalue of the selfadjoint operator
%
\begin{math}
  (p-1)^{-1} \mathcal{A}_{\varphi_1} + \lambda_1\varphi_1^{p-2} m
    \colon
  \mathcal{D}_{\varphi_1} \to \mathcal{D}_{\varphi_1}' ,
\end{math}
%
where
$\mathcal{A}_{\varphi_1}$ has been defined in~\eqref{def.A}.
The following result determines all minimizers
for~\eqref{eq.lam_1}:

%
%%%%%    UNIQUENESS and POSITIVITY - EV (Proposition)
%
\begin{proposition}\label{prop-Uni-EV}
Let\/ $1<p<N$ and let hypothesis \eqref{hyp:m>0} be satisfied.
Then a function $u\in \mathcal{D}_{\varphi_1}$ satisfies
$\mathcal{Q}_0 (u,u) = 0$ if and only if
$u = \kappa\varphi_1$ for some constant $\kappa\in \mathbb{R}$.
%
\end{proposition}
%

The analogue of this proposition for a bounded domain
$\Omega\subset \mathbb{R}^N$ with a sufficiently regular boundary
$\partial\Omega$ is due to
{\sc Tak\'a\v{c}} \cite[Prop.\ 4.4, p.~202]{Takac-1}.
Our proof of Proposition~\ref{prop-Uni-EV} below
is a simplification of that given in \cite{Takac-1}. \smallskip


\begin{proof}{Proof of Proposition~\ref{prop-Uni-EV}}
Recall that the embedding
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
is compact, by
Proposition \ref{prop-Compact}(b).
Let $u$ be any (nontrivial) minimizer for $\lambda_1$
in~\eqref{eq.lam_1}.
If $u$ changes sign in $\mathbb{R}^N$, denote
$u^+ = \max\{ u, 0\}$ and $u^- = \max\{ -u, 0\}$.
Then we have, using
{\sc Gilbarg} and {\sc Trudinger}
\cite[Theorem 7.8, p.~153]{GilbargTrud},
%
\begin{align*}
  \lambda_1
& =
  \frac{ \int_{\mathbb{R}^N} (u^+)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
       { \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
    \cdot
  \frac{ \int_{\mathbb{R}^N}
    \langle {\bf A}(\nabla\varphi_1) \nabla u^+, \nabla u^+
    \rangle_{\mathbb{R}^N} \,{\rm d}x }%
       { (p-1) \int_{\mathbb{R}^N} (u^+)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\\
&\quad +
  \frac{ \int_{\mathbb{R}^N} (u^-)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
       { \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
    \cdot
  \frac{ \int_{\mathbb{R}^N}
    \langle {\bf A}(\nabla\varphi_1) \nabla u^-, \nabla u^-
    \rangle_{\mathbb{R}^N} \,{\rm d}x }%
       { (p-1) \int_{\mathbb{R}^N} (u^-)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\\
& \geq
    \Big(
    \frac{ \int_{\mathbb{R}^N} (u^+)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
         { \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
  + \frac{ \int_{\mathbb{R}^N} (u^-)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
         { \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
    \Big) \lambda_1
  = \lambda_1 .
\end{align*}
%
Consequently, both
$u^+$ and $u^-$ are (nontrivial) minimizers for $\lambda_1$.

Next, we show that if $u\in \ker(\mathcal{A}_{\varphi_1})$
then $u$ is a constant multiple of $\varphi_1$.
Since
$\varphi_1$ satisfies~\eqref{e:varphi.l_1},
it is of class $C^\infty$ in $\mathbb{R}^N\setminus \{ 0\}$,
by classical regularity theory
\cite[Theorem 8.10, p.~186]{GilbargTrud}.
Now, for each $\gamma\in \mathbb{R}$ fixed, consider the function
$v_\gamma\eqdef u - \gamma\varphi_1$ in $\mathbb{R}^N$.
Then both $v_\gamma^+$ and $v_\gamma^-$ belong to
$\ker(\mathcal{A}_{\varphi_1})$ and thus satisfy the equation
%
\begin{equation}
  - \nabla\cdot
    \left( {\bf A}(\nabla\varphi_1) \nabla v_\gamma^\pm \right)
  = \lambda_1 (p-1) \varphi_1^{p-2} m v_\gamma^\pm \geq 0
    \quad\mbox{in } \mathbb{R}^N\setminus \{ 0\} .
\label{A_phi=0:U}
\end{equation}
%
Again, we have
$v_\gamma^\pm \in C^\infty( \mathbb{R}^N\setminus \{ 0\} )$.
So we may apply the strong maximum principle
\cite[Theorem 3.5, p.~35]{GilbargTrud}
to eq.~\eqref{A_phi=0:U}
to conclude that either
$v_\gamma^+ \equiv 0$ in $\mathbb{R}^N\setminus \{ 0\}$, or else
$v_\gamma^+ > 0$ throughout $\mathbb{R}^N\setminus \{ 0\}$,
and similarly for $v_\gamma^-$.
This means that
$\mathop{\rm sign} (u - \gamma\varphi_1) \equiv \mathrm{const}$
in $\mathbb{R}^N\setminus \{ 0\}$.
Moving $\gamma$ from $-\infty$ to $+\infty$, we get
$u\equiv \kappa\varphi_1$ in $\mathbb{R}^N\setminus \{ 0\}$
for some constant $\kappa\in \mathbb{R}$.
This means
$u = \kappa\varphi_1$ in $\mathcal{D}_{\varphi_1}$, as claimed.
\end{proof}


\section{An improved Poincar\'e inequality ($2\leq p < N$)}
\label{s:Impr_Poinc}

We need a few more technical tools from
{\sc Fleckinger} and {\sc Tak\'a\v{c}} \cite[Sect.~5]{FleckTakac-1}
to prove Lemma~\ref{lem-Poincare}.
Although our present situation requires only a few changes
in the space setting in \cite{FleckTakac-1},
we provide complete proofs of all results
for the convenience of the reader.

%
%%%%%    An Improved Poincare INEQUALITY (Remark)
%
\begin{remark}\label{rem-Poincare}
\begingroup\rm
Except when $u^\parallel = 0$, we may replace
$u\in D^{1,p}(\mathbb{R}^N)$ by $v = u / u^\parallel$
in inequality~\eqref{e:Poincare}
and thus restate it equivalently as follows, for all
$v^\top\in D^{1,p}(\mathbb{R}^N)$ with
$\int_{\mathbb{R}^N} v^\top\, \varphi_1^{p-1}\, m \,{\rm d}x = 0$:
%
\begin{equation}
    \mathcal{Q}_{v^\top}(v^\top, v^\top)
  = \mathcal{F}(\varphi_1 + v^\top)
  \geq \frac{c}{p}
    \left(
      \| v^\top\|_{ \mathcal{D}_{\varphi_1} }^2
    + \| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }^p
    \right) .
\label{e:Poinc}
\end{equation}
%
\endgroup
\end{remark}
%


This remark indicates that
our proof of inequality~\eqref{e:Poincare}
should distinguish between the cases when the ratio
$\| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } / | u^\parallel |$
is bounded away from zero by a constant $\gamma > 0$, say,
\[
  \| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } / | u^\parallel |
  \geq \gamma ,
\]
and when it is sufficiently small, say,
\[
  \| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } / | u^\parallel |
  \leq \gamma
\]
where $\gamma > 0$ is small enough.
The former case is treated in a standard way analogous
to~\eqref{def.lam_1},
whereas
the latter case requires a more sophisticated approach based on
the second\--order Taylor formula \eqref{def.Q}
applied to the expression
$\mathcal{Q}_{v^\top}(v^\top, v^\top)$
on the left\--hand side in~\eqref{e:Poinc}
where $v = u / u^\parallel$.
For either of these cases we need a separate auxiliary result:
We derive two formulas for Rayleigh quotients
outside and inside an arbitrarily small cone around
the axis spanned by $\varphi_1$, respectively.

%
\subsection{Minimization outside a cone around $\varphi_1$}
\label{ss:Ext-Cone}

We allow $1 < p < N$ throughout this paragraph.
Given any number $0 < \gamma < \infty$, we set
%
\begin{gather*}
 \mathcal{C}_\gamma \eqdef
    \Big\{
  u\in D^{1,p}(\mathbb{R}^N) \colon
  \| u^\top \|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma | u^\parallel |
    \Big\} ,\\
 \mathcal{C}_\gamma^\prime \eqdef
    \Big\{
  u\in D^{1,p}(\mathbb{R}^N) \colon
  \| u^\top \|_{ D^{1,p}(\mathbb{R}^N) } \geq \gamma | u^\parallel |
    \Big\} .
\end{gather*}
%
Note that $\mathcal{C}_\gamma$ is a closed cone in $D^{1,p}(\mathbb{R}^N)$ and
$\mathcal{C}_\gamma^\prime$ is the closure of $\mathcal{C}_\gamma^c$,
the complement of $\mathcal{C}_\gamma$ in $D^{1,p}(\mathbb{R}^N)$.
We consider also the hyperplane
\[
  \mathcal{C}_\infty^\prime \eqdef
    \Big\{
  u\in D^{1,p}(\mathbb{R}^N) \colon u^\parallel = 0
    \Big\}
  = \bigcap_{ 0 < \gamma < \infty } \mathcal{C}_\gamma^\prime .
\]
For $0 < \gamma\leq \infty$ we define
%
\begin{equation}
  \Lambda_\gamma \eqdef
    \inf \Big\{
  \frac{ \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x }%
       { \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x }
  \colon u\in \mathcal{C}_\gamma^\prime \setminus \{ 0\}
    \Big\} .
\label{def.Lam_g}
\end{equation}
%

The next result is an analogue of
\cite[Lemma 5.1, p.~963]{FleckTakac-1}
proved for a bounded domain $\Omega\subset \mathbb{R}^N$.


%%%%%    Exterior Cone Minimum (Lemma)

\begin{lemma}\label{lem-Ext-Cone}
Let\/ $1 < p < N$ and\/ $0 < \gamma\leq \infty$.
Then we have $\Lambda_\gamma > \lambda_1$.
\end{lemma}
%

\begin{proof}
Assume the contrary, that is,
$\Lambda_\gamma = \lambda_1$ for some $0 < \gamma < \infty$.
Pick a minimizing sequence
$\{ u_n\}_{n=1}^\infty$ in $\mathcal{C}_\gamma^\prime$ such that
\[
  \int_{\mathbb{R}^N} |u_n|^p\, m \,{\rm d}x = 1
    \quad\mbox{and }\quad
  \int_{\mathbb{R}^N} |\nabla u_n|^p \,{\rm d}x \to \lambda_1
  \quad\mbox{as } n\to \infty .
\]
Since $D^{1,p}(\mathbb{R}^N)$ is a reflexive Banach space,
the minimizing sequence contains a weakly convergent subsequence in
$D^{1,p}(\mathbb{R}^N)$ which we denote by
$\{ u_n\}_{n=1}^\infty$ again.
Consequently,
$u_n\to u$ strongly in $L^p(\mathbb{R}^N; m)$, by
Proposition \ref{prop-Compact}(a), and
$\nabla u_n\rightharpoonup \nabla u$ weakly in $[ L^p(\mathbb{R}^N) ]^N$
as $n\to \infty$.
We deduce that
$\int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x = 1$ and
\[
  \lambda_1^{1/p} \leq
  \| \nabla u\|_{L^p(\mathbb{R}^N)} \leq
  \liminf_{n\to \infty} \| \nabla u_n\|_{L^p(\mathbb{R}^N)}
  = \lambda_1^{1/p} .
\]
As the standard norm on the space $D^{1,p}(\mathbb{R}^N)$
is uniformly convex,
by Clarkson's inequalities, we must have
$u_n\to u$ strongly in $D^{1,p}(\mathbb{R}^N)$,
by the proof of Milman's theorem
(see {\sc Yosida} \cite[Theorem V.2.2, p.~127]{Yosida}).
This means that
%
\begin{gather*}
   u_n^\parallel
  = \int_{\mathbb{R}^N} u_n\, \varphi_1^{p-1}\, m \,{\rm d}x
  \to
    u^\parallel
  = \int_{\mathbb{R}^N} u\, \varphi_1^{p-1}\, m \,{\rm d}x , \\
   u_n^\top = u_n - u_n^\parallel \varphi_1 \to
      u^\top =   u -   u^\parallel \varphi_1
    \mbox{ strongly in } D^{1,p}(\mathbb{R}^N) ,
\end{gather*}
%
as $n\to \infty$.
The set $\mathcal{C}_\gamma^\prime$ being closed in $D^{1,p}(\mathbb{R}^N)$,
we thus have
$u\in \mathcal{C}_\gamma^\prime$.

On the other hand, from
$\| u\|_{ L^p(\mathbb{R}^N; m) } = 1$ and
$\| \nabla u\|_{L^p(\mathbb{R}^N)} = \lambda_1^{1/p}$,
combined with the simplicity of the first eigenvalue $\lambda_1$,
one deduces that $u = \pm\varphi_1$, a contradiction to
$u\in \mathcal{C}_\gamma^\prime$.
The lemma is proved.
\end{proof}

%
\subsection{Minimization inside a cone around $\varphi_1$}
\label{ss:Int-Cone}

For $\phi\in D^{1,p}(\mathbb{R}^N)$, $\phi\not\equiv 0$ in $\mathbb{R}^N$,
let us define
%
\begin{equation}
  \tilde\Lambda \eqdef
    \liminf_{
  \begin{smallmatrix}
                     { \|\phi\|_{ D^{1,p}(\mathbb{R}^N) } \to 0 }\\
                     { \langle\phi, \varphi_1^{p-1} m\rangle = 0 }
  \end{smallmatrix}
  }
  \frac{
    \int_{\mathbb{R}^N}
    \left\langle
  \left[
    \int_0^1 \mathbf{A}
  \left(
    \nabla ( \varphi_1 + s\phi )
  \right) (1-s) \,{\rm d}s
  \right] \nabla\phi ,\, \nabla\phi
    \right\rangle_{\mathbb{R}^N} \,{\rm d}x
  }{
    \int_{\mathbb{R}^N}
    \left[
    \int_0^1 |\varphi_1 + s\phi|^{p-2} (1-s) \,{\rm d}s
    \right] |\phi|^2\, m \,{\rm d}x
  }
\label{def.Lam_0}
\end{equation}
%
with the abbreviation~\eqref{def.A=F''}.
Using the quadratic form $\mathcal{Q}_\phi$ defined
in~\eqref{def.Q}, we notice that
\[
  \tilde\Lambda - \lambda_1 (p-1) =
    \liminf_{
  \begin{smallmatrix}
                     { \|\phi\|_{ D^{1,p}(\mathbb{R}^N) } \to 0 }\\
                     { \langle\phi, \varphi_1^{p-1} m\rangle = 0 }
  \end{smallmatrix}
  }
  \frac{ \mathcal{Q}_{\phi}(\phi,\phi)
  }{
    \int_{\mathbb{R}^N}
    \left[
    \int_0^1 |\varphi_1 + s\phi|^{p-2} (1-s) \,{\rm d}s
    \right] |\phi|^2\, m \,{\rm d}x
  } \geq 0 .
\]

The next result parallels
\cite[Lemma 5.2, p.~964]{FleckTakac-1}
shown for a bounded domain $\Omega\subset \mathbb{R}^N$.

%%%%%    Interior Cone Minimum (Lemma)

\begin{lemma}\label{lem-Int-Cone}
Let\/ $2\leq p < N$.
We have $\tilde\Lambda > \lambda_1 (p-1)$.
%
\end{lemma}
%

Before giving the proof of this inequality,
we first recall that the kernels of the quadratic forms
$\mathcal{Q}_{\phi}(v,v)$ and
$\mathcal{Q}_0(v,v)$ defined in
\eqref{def.Q} and \eqref{def.Q_0}, respectively,
can be compared by inequalities
\eqref{1-s.A.geom:p>2} for $p\geq 2$, and
\eqref{1-s.A.geom:p<2} for $p<2$,
so that we can use the Hilbert space $\mathcal{D}_{\varphi_1}$
not only for $\mathcal{Q}_0$ but also for $\mathcal{Q}_{\phi}$.

Next, we introduce the following notations where
$t\in \mathbb{R}$ and $\phi\in D^{1,p}(\mathbb{R}^N)$:
%
\begin{gather*}
 \mathcal{P}_0(t,\phi) \eqdef
    \int_{\mathbb{R}^N}
    \Big[ \int_0^1
    | \varphi_1 + s t\phi |^{p-2} (1-s) \,{\rm d}s
    \Big] \phi^2\, m \,{\rm d}x ,\\
 \mathcal{P}_1(t,\phi) \eqdef
    \int_{\mathbb{R}^N}
    \Big\langle
  \Big[
    \int_0^1 \mathbf{A}
  (
    \nabla ( \varphi_1 + s t\phi )
  ) (1-s) \,{\rm d}s
  \Big] \nabla\phi ,\, \nabla\phi
    \Big\rangle_{\mathbb{R}^N} \,{\rm d}x .
\end{gather*}
%
Hence, equation~\eqref{def.Lam_0} takes the form
\[
  \tilde\Lambda =
    \liminf_{
  \begin{smallmatrix}
                     { \|\phi\|_{ D^{1,p}(\mathbb{R}^N) } \to 0 }\\
                     { \langle\phi, \varphi_1^{p-1} m\rangle = 0 }
  \end{smallmatrix}
  }
  \frac{ \mathcal{P}_1(t,\phi) }{ \mathcal{P}_0(t,\phi) }
    \quad\mbox{with any fixed } t\in \mathbb{R}\setminus \{ 0\} .
\]
Furthermore, due to inequalities \eqref{1-s.A.geom:p>2},
the expressions
$\mathcal{P}_0(t,\phi)$ and $\mathcal{P}_1(t,\phi)$,
respectively, are equivalent to
%
\begin{align*}
  \mathcal{N}_0(t,\phi) &\eqdef
    \int_{\mathbb{R}^N}
    \left(
      \varphi_1^{p-2} + |t|^{p-2} |\phi|^{p-2}
    \right) \phi^2\, m \,{\rm d}x
\\
& = \int_{\mathbb{R}^N} \varphi_1^{p-2} \phi^2\, m \,{\rm d}x
  + |t|^{p-2} \|\phi\|_{ L^p(\mathbb{R}^N; m) }^p
\end{align*}
%
and
%
\begin{align*}
  \mathcal{N}_1(t,\phi) &\eqdef
    \int_{\mathbb{R}^N}
    \left(
      |\nabla\varphi_1|^{p-2}
    + |t|^{p-2} |\nabla\phi|^{p-2}
    \right) |\nabla\phi|^2 \,{\rm d}x
\\
& = \|\phi\|_{ \mathcal{D}_{\varphi_1} }^2
  + |t|^{p-2} \|\phi\|_{ D^{1,p}(\mathbb{R}^N) }^p ,
\end{align*}
%
that is, there are two constants
$c_1, c_2 > 0$ independent from $t$ and $\phi$ such that
%
\begin{equation}
  c_1\, \mathcal{N}_i(t,\phi) \leq \mathcal{P}_i(t,\phi) \leq
  c_2\, \mathcal{N}_i(t,\phi) ;\quad i=0,1 .
\label{ineq.PN}
\end{equation}
%

\begin{proof}[Proof of Lemma~\ref{lem-Int-Cone}]
On the contrary, assume that
$\tilde\Lambda \leq \lambda_1 (p-1)$.
Pick a minimizing sequence
$\{ \phi_n\}_{n=1}^\infty$ in $D^{1,p}(\mathbb{R}^N)$
such that
$\phi_n\not\equiv 0$ in $\mathbb{R}^N$,
$\langle\phi_n, \varphi_1^{p-1} m\rangle = 0$,
$\|\phi_n\|_{ D^{1,p}(\mathbb{R}^N) } \to 0$,
and
\[
  \frac{ \mathcal{P}_1(1,\phi_n) }{ \mathcal{P}_0(1,\phi_n) }
  \,\longrightarrow \tilde\Lambda \leq \lambda_1 (p-1)
    \quad\mbox{as } n\to \infty .
\]
Next, set
$t_n = \mathcal{P}_0(1,\phi_n)^{1/2}$
and
$V_n = \phi_n / t_n$ for $n=1,2,\dots$.
Hence, we have \hbox{$t_n\to 0$},
$\mathcal{P}_0(t_n,V_n) = 1$, and
$\mathcal{P}_1(t_n,V_n) \to \tilde\Lambda$ as $n\to \infty$.
Inequalities~\eqref{ineq.PN} guarantee that both sequences
$\| V_n\|_{ \mathcal{D}_{\varphi_1} }$ and
$t_n^{1 - (2/p)} \| V_n\|_{ D^{1,p}(\mathbb{R}^N) }$
are bounded, and so we may extract a subsequence
denoted again by $\{ V_n\}_{n=1}^\infty$ such that
$V_n\rightharpoonup V$ weakly in $\mathcal{D}_{\varphi_1}$ and
$t_n^{1 - (2/p)} V_n$ $\rightharpoonup z$ weakly in
$D^{1,p}(\mathbb{R}^N)$ as $n\to \infty$.
Using the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$,
we get $z\equiv 0$ in $\mathbb{R}^N$.
Furthermore, both imbeddings
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^p(\mathbb{R}^N; m)$ and
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
being compact by Proposition \ref{prop-Compact},
we have also
$V_n\to V$ strongly in $\mathcal{H}_{\varphi_1}$ and
$t_n^{1 - (2/p)} V_n\to 0$ strongly in $L^p(\mathbb{R}^N; m)$.
It follows that
$\langle V, \varphi_1^{p-1} m\rangle = 0$ and
%
\begin{gather*}
 \mathcal{P}_0(0,V) = \frac12
    \int_{\mathbb{R}^N} \varphi_1^{p-2} V^2 \,{\rm d}x = 1 , \\
 \mathcal{P}_1(0,V) = \frac12
    \left\langle
  \mathbf{A} (\nabla\varphi_1) \nabla V ,\, \nabla V
    \right\rangle \leq \tilde\Lambda \leq \lambda_1 (p-1) .
\end{gather*}
%
Consequently, Proposition~\ref{prop-Uni-EV} forces
$V = \kappa\varphi_1$ in $\mathbb{R}^N$,
where $\kappa\in \mathbb{R}$ is a constant,
$\kappa\neq 0$ by $\mathcal{P}_0(0,V) = 1$.
But this is a contradiction to
$\langle V, \varphi_1^{p-1} m\rangle = 0$.
We conclude that
$\tilde\Lambda > \lambda_1 (p-1)$ as claimed.
\end{proof}

\subsection{Proof of Lemma~\ref{lem-Poincare}}
\label{ss:Poincare}

If $u\in D^{1,p}(\mathbb{R}^N)$ satisfies
$\langle u, \varphi_1\rangle = 0$,
then equation~\eqref{def.Lam_g} implies
%
\begin{equation}
\begin{split}
  \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
  - \lambda_1 \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x
& \geq
    \big( 1 - \frac{\lambda_1}{\Lambda_\infty} \big)
    \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x\\
&= \big( 1 - \frac{\lambda_1}{\Lambda_\infty} \big)
    \int_{\mathbb{R}^N} |\nabla u^\top|^p \,{\rm d}x
\end{split}
\label{ineq:Lam_inf}
\end{equation}
%
where $\lambda_1 / \Lambda_\infty < 1$
by Lemma~\ref{lem-Ext-Cone}.
Thus, we may assume
$\langle u, \varphi_1\rangle \not= 0$
and so we need to prove only inequality~\eqref{e:Poinc}.
We will apply Lemmas \ref{lem-Ext-Cone} and~\ref{lem-Int-Cone}
to the following two cases, respectively.

\noindent {\it Case\/}
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \geq \gamma$:
Here, $\gamma > 0$ is an arbitrary, but fixed number.
In analogy with inequality~\eqref{ineq:Lam_inf} above,
we have
%
\begin{equation}
\begin{split}
&   \int_{\mathbb{R}^N} |\nabla\varphi_1 + \nabla v^\top|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |\varphi_1 + v^\top|^p\, m \,{\rm d}x
\\
& \geq
    \big( 1 - \frac{\lambda_1}{\Lambda_\gamma} \big)
    \int_{\mathbb{R}^N} |\nabla\varphi_1 + \nabla v^\top|^p \,{\rm d}x
  \geq c_\gamma
    \int_{\mathbb{R}^N} |\nabla v^\top|^p \,{\rm d}x
\end{split}
\label{ineq:Lam_g}
\end{equation}
%
for all $v^\top\in D^{1,p}(\mathbb{R}^N)$ such that
$\langle v^\top, \varphi_1^{p-1} m\rangle = 0$
and
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \geq \gamma$,
where $c_\gamma > 0$ is a constant independent from $v^\top$.
The last inequality follows from the boundedness of
the orthogonal projections
$u\mapsto u^\parallel\cdot \varphi_1$ and
$u\mapsto u^\top$ in $D^{1,p}(\mathbb{R}^N)$.
Recalling the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$,
we deduce from~\eqref{ineq:Lam_g}
that inequality~\eqref{e:Poinc} is valid provided
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }$ $\geq \gamma$.

\noindent {\it Case\/}
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma$:
Here, $\gamma > 0$ is sufficiently small.
According to equation~\eqref{def.Lam_0} and
Lemma~\ref{lem-Int-Cone} we have
%
\begin{equation}
\begin{split}
   \mathcal{Q}_{v^\top}(v^\top, v^\top)
&= \mathcal{P}_1(1,v^\top)
  - \lambda_1 (p-1)\, \mathcal{P}_0(1,v^\top)
\\
& \geq
    \Big( 1 - \frac{\lambda_1 (p-1)}{\tilde\Lambda}
    \Big)
    \mathcal{P}_1(1,v^\top)\\
&\geq \tilde c\cdot \mathcal{N}_1(1,v^\top)
\end{split}
\label{ineq:Lam_0}
\end{equation}
%
for all $v^\top\in D^{1,p}(\mathbb{R}^N)$ such that
$\langle v^\top, \varphi_1^{p-1} m\rangle  = 0$
and
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma$,
where $\gamma > 0$ is sufficiently small and
$\tilde c > 0$ is a constant independent from $v^\top$.
Recall that the expressions
$\mathcal{P}_i(1,v^\top)$ and
$\mathcal{N}_i(1,v^\top)$ ($i=0,1$)
have been defined after Lemma~\ref{lem-Int-Cone}.
From~\eqref{ineq:Lam_0} we deduce that
inequality~\eqref{e:Poinc} is valid also when
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma$.
%\end{proof}


%%%%%    Coercivity of J_{\lambda_1} (Remark)

\begin{remark}\label{rem-E_f:coercive}
\begingroup\rm
Assume $2 < p < N$ and let
$f\in \mathcal{D}_{\varphi_1}'$ satisfy
$\langle f,\varphi_1\rangle = 0$.
Recall that
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$.
Although the functional $\mathcal{J}_{\lambda_1}$,
defined in \eqref{def.jl} with $\lambda = \lambda_1$,
is no longer coercive on $D^{1,p}(\mathbb{R}^N)$,
it is still not only bounded from below, but also
``very close'' to being coercive on the weighted Sobolev space
$\mathcal{D}_{\varphi_1}$,
as a direct consequence of improved Poincar\'e's inequality
\eqref{e:Poincare}.
This property of $\mathcal{J}_{\lambda_1}$
will be used in the next section to prove the existence theorem
(Theorem~\ref{thm-Exist:p>2})
for problem \eqref{e:BVP.l_1}.
\endgroup
\end{remark}
%

\section{Proof of Theorem~\ref{thm-Exist:p>2}}
\label{s:pr-Exist:p>2}

Our proof of Theorem~\ref{thm-Exist:p>2}
combines the improved Poincar\'e inequality~\eqref{e:Poincare}
with a generalized Rayleigh quotient formula.
To this end, we may assume that
$f\in \mathcal{D}_{\varphi_1}'$ satisfies
$f\not\equiv 0$ in $\mathbb{R}^N$ and
$\langle f, \varphi_1\rangle = 0$.
Define the number $M_f$, for $0\leq M_f\leq \infty$, by
%
\begin{equation}
  M_f\eqdef
    \sup_{
  \begin{smallmatrix}
    v\in D^{1,p}(\mathbb{R}^N)\\
    v\not\in \{ \kappa\varphi_1\colon \kappa\in \mathbb{R}\}
  \end{smallmatrix}
  }
  \frac{ | \langle f, v\rangle |^p }%
  {
    \int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
  }\, .
\label{def.M_f}
\end{equation}
%
Clearly, $M_f > 0$.
Moreover,
inequality~\eqref{e:Poincare} entails
%
\[
   | \langle f, v\rangle |^p
  \leq \| f\|_{ D^{-1,p'}(\mathbb{R}^N) }^p\,
       \| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }^p
  \leq C_f
    \Big(
    \int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
    \Big)
\]
%
for all $v\in D^{1,p}(\mathbb{R}^N)$, where
$C_f = c^{-1}\, \| f\|_{ D^{-1,p'}(\mathbb{R}^N) }^p$
is a constant.
This shows that $M_f\leq C_f < \infty$.
In a similar way we arrive at
%
\begin{equation}
\begin{split}
  | v^\parallel |^{p-2}\,
    | \langle f, v\rangle |^2
&\leq | v^\parallel |^{p-2}
       \left( \| f\|_{ \mathcal{D}_{\varphi_1}' } \right)^2\,
       \| v^\top\|_{ \mathcal{D}_{\varphi_1} }^2 \\
& \leq C_f'
    \Big(
    \int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
    \Big)
  \quad\mbox{for all } v\in D^{1,p}(\mathbb{R}^N) ,
\end{split}
\label{ineq:f.v_2}
\end{equation}
%
where
$C_f'= c^{-1} ( \| f\|_{ \mathcal{D}_{\varphi_1}' } )^2$
is a constant, and
$\|\cdot\|_{ \mathcal{D}_{\varphi_1}' }$
stands for the dual norm on $\mathcal{D}_{\varphi_1}'$.
>From \eqref{def.M_f} and inequality~\eqref{ineq:f.v_2}
we can draw the following conclusion:
If $v\in D^{1,p}(\mathbb{R}^N)$ is such that
$v^\top \not\equiv 0$ in $\mathbb{R}^N$ and
\[
  \frac{ | \langle f, v\rangle |^p }%
  {
    \int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
  }
  \geq \frac{1}{2} M_f ,
\]
then
$\langle f, v\rangle \neq 0$ and
\[
    | v^\parallel |^{p-2}
  \leq 2 (C_f' / M_f)\,
    | \langle f, v\rangle |^{p-2}
  \leq (C_f'')^{p-2}\,
    \| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }^{p-2} ,
\]
where
%
\begin{math}
  C_f''=
  [ 2 (C_f' / M_f) ]^{1/(p-2)}\, \| f\|_{ D^{-1,p'}(\mathbb{R}^N) }
\end{math}
%
is a constant, i.e.,
%
\begin{equation}
    | v^\parallel |
  \leq C_f''\,
    \| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } .
\label{ineq:v_f.v}
\end{equation}
%

Next, take any maximizing sequence
$\{ v_n\}_{n=1}^\infty$ in $D^{1,p}(\mathbb{R}^N)$
for the generalized Rayleigh quotient~\eqref{def.M_f},
that is, $v_n^\top \not\equiv 0$ in $\mathbb{R}^N$ and
%
\begin{equation}
  \frac{ | \langle f, v_n\rangle |^p }%
  {
    \int_{\mathbb{R}^N} |\nabla v_n|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |v_n|^p\, m \,{\rm d}x
  }\,\longrightarrow M_f
    \quad\mbox{as } n\to \infty .
\label{M_f:v_n}
\end{equation}
%
Since both, the numerator and the denominator are $p$-homogeneous,
we may assume
$\| v_n\|_{ D^{1,p}(\mathbb{R}^N) } = 1$ for all $n\geq 1$.
The Sobolev space $D^{1,p}(\mathbb{R}^N)$ being reflexive,
we may pass to a convergent subsequence
$v_n\rightharpoonup w$ weakly in $D^{1,p}(\mathbb{R}^N)$;
hence, also $v_n\to w$ strongly in $L^p(\mathbb{R}^N; m)$, by
Proposition \ref{prop-Compact}(a), and
$\langle f, v_n\rangle \to \langle f, w\rangle$
as $n\to \infty$.
We insert these limits into~\eqref{M_f:v_n} to obtain
%
\begin{equation}
    \int_{\mathbb{R}^N} |\nabla w|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |w|^p\, m \,{\rm d}x
  \leq 1 - \lambda_1
    \int_{\mathbb{R}^N} |w|^p\, m \,{\rm d}x
  = M_f^{-1}
    | \langle f, w\rangle |^p .
\label{M_f:w}
\end{equation}
%
In particular, we have $w\not\equiv 0$ in $\mathbb{R}^N$,
therefore also $w^\top \not\equiv 0$
by~\eqref{ineq:v_f.v}, and consequently
$| \langle f, w\rangle | \not= 0$
by~\eqref{M_f:w}.
We combine \eqref{def.M_f} with~\eqref{M_f:w} to get
$\int_{\mathbb{R}^N} |\nabla w|^p \,{\rm d}x = 1$.
Hence, the supremum $M_f$ in~\eqref{def.M_f}
is attained at $w$ in place of $v$.

Finally, we can apply
the calculus of variations to the inequality
%
\[
   \int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
  - \lambda_1
    \int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
  - M_f^{-1}
    | \langle f, v\rangle |^p
  \geq 0
    \quad\mbox{for } v\in D^{1,p}(\mathbb{R}^N)
\]
%
to derive
%
\[
  - \Delta_p w - \lambda_1\, m\, |w|^{p-2} w
  = M_f^{-1}
    | \langle f, w\rangle |^{p-2}
      \langle f, w\rangle \cdot f(x)
    \quad\mbox{in } \mathbb{R}^N .
\]
%
It follows that
$u\eqdef M_f^{1/(p-1)} \langle f, w\rangle^{-1} \cdot w$
is a weak solution of problem~\eqref{e:BVP.l_1}.
Theorem~\ref{thm-Exist:p>2} is proved.
%\end{proof}

\section{Proof of Theorem~\ref{thm-Exist:p<2}}
\label{s:pr-Exist:p<2}

In contrast to the case $2\leq p < N$ in Section~\ref{s:Impr_Poinc},
Remark~\ref{rem-E_f:coercive},
for $1<p<2$ the functional $\mathcal{J}_{\lambda_1}$
will turn out to be unbounded from below on $D^{1,p}(\mathbb{R}^N)$
along curves ``close'' to $\pm\tau \varphi_1$ as $\tau\to +\infty$,
even though it still remains coercive on the complement
$D^{1,p}(\mathbb{R}^N)^\top$ of $\mathop{\rm lin} \{ \varphi_1\}$ in
$D^{1,p}(\mathbb{R}^N)$ defined in \eqref{def.D^top}.
Again, we take advantage of the direct sum
%
\begin{math}
  D^{1,p}(\mathbb{R}^N) =
  \mathop{\rm lin} \{ \varphi_1\} \oplus D^{1,p}(\mathbb{R}^N)^\top
\end{math}
%
defined in~\eqref{ortho:u}.
These facts show that the functional $\mathcal{J}_{\lambda_1}$
has a simple saddle point geometry.
Such a scenario is typically suitable for a saddle point theorem
which guarantees the existence of a critical point for
$\mathcal{J}_{\lambda_1}$ by means of a minimax formula for
a critical value of $\mathcal{J}_{\lambda_1}$.
Here we make use of a ``very direct'' minimax method introduced in
{\sc Tak\'a\v{c}} \cite[Sect.~7]{Takac-1}
which does not require any Palais\--Smale condition.
In this section we adapt this method to our setting.
In a closely related work
{\sc Dr\'abek} and {\sc Holubov\'a} \cite[Theorem 1.1]{DrabHolub}
applied the saddle point theorem from
{\sc Rabinowitz} \cite[Theorem 4.6, p.~24]{Rabin}
to establish an existence result for problem \eqref{e:BVP.l_1} when
$\Omega\subset \mathbb{R}^N$ is a bounded domain and $1<p<2$.

As we allow the function $f\in D^{-1,p'}(\mathbb{R}^N)$ in
the energy functional \eqref{def.jl} to vary, we write
$\mathcal{J}_{\lambda}(u) \equiv \mathcal{J}_{\lambda}(u;f)$.

%
\subsection{Simple saddle point geometry}
\label{ss:Saddle_Geom}

For $\lambda = \lambda_1$ it will be convenient to use the notation
%
\begin{equation*}
  \mathcal{E}_f(u) \eqdef \mathcal{J}_{\lambda_1}(u;f)
    \quad\mbox{for } u\in D^{1,p}(\mathbb{R}^N) .
\end{equation*}
%

\begin{proof}[Proof of Lemma~\ref{lem-Saddle_Geom}]
We infer from Lemma~\ref{lem-Ext-Cone} that
$\Lambda_\infty > \lambda_1$ in formula \eqref{def.Lam_g}.
This shows that the functional $\mathcal{E}_f$ is coercive on
$\mathcal{C}_\infty^\prime = D^{1,p}(\mathbb{R}^N)^\top$.
Hence, being also weakly lower semicontinuous,
$\mathcal{E}_f$ possesses a global minimizer $u_0^\top$ over
$D^{1,p}(\mathbb{R}^N)^\top$,
\[
    \mathcal{E}_f(u_0^\top)
  = \inf_{ w\in D^{1,p}(\mathbb{R}^N)^\top } \mathcal{E}_f(w)
  > -\infty .
\]

Now let us look for the functions $u$ and $v$, respectively,
in Definition \ref{def-Saddle_Geom} in the forms of
%
\begin{equation}
  u_\pm = \pm\tau \varphi_1 + \tau^{1 - (p/2)} \phi
  \quad\mbox{with $\tau\in (0,\infty)$ sufficiently large, }
\label{def.u,v}
\end{equation}
%
where $\phi\in C_{\mathrm{c}}^1(\mathbb{R}^N)$
is a function chosen as follows:
%
\begin{itemize}
%
\item[($\boldsymbol{\Phi}$)]
$\langle f, \phi\rangle = 1$ and $0\not\in K$ where
%
\begin{equation*}
  K = \mathop{\rm supp}(\phi)\eqdef
      \overline{ \{ x\in \mathbb{R}^N\colon \phi(x)\not= 0\} }
    \quad ( \subset \mathbb{R}^N )
\end{equation*}
%
denotes the support of $\phi$.
%
\end{itemize}
%

The existence of $\phi$ is verified as follows.
Since
$f\in D^{-1,p'}(\mathbb{R}^N)$ satisfies
$f\not\equiv 0$ in $\mathbb{R}^N$, there is a function
$\phi_0\in C_{\mathrm{c}}^1(\mathbb{R}^N)$ such that
$\langle f, \phi_0\rangle = 1$.
On the contrary to ($\boldsymbol{\Phi}$),
suppose that the support
$K_0 = \mathop{\rm supp}(\phi_0)$ of $\phi_0$ always contains $0\in \mathbb{R}^N$.
This is equivalent to saying that
$\langle f, \phi\rangle = 0$ whenever
$\phi\in C_{\mathrm{c}}^1(\mathbb{R}^N)$ is such that
$0\not\in \mathop{\rm supp}(\phi)$.
Now choose a $C^1$ function
$\psi\colon \mathbb{R}_+\to [0,1]$ such that
$\psi(r) = 1$ if $0\leq r\leq 1$,
$0\leq \psi(r)\leq 1$ if $1\leq r\leq 2$, and
$\psi(r) = 0$ if $2\leq r < \infty$.
Define
$\psi_n(x)\eqdef \psi(n|x|)$ for all $x\in \mathbb{R}^N$; $n=1,2,\dots$.
Then
$0\not\in \mathop{\rm supp}( (1-\psi_n)\phi_0 )$ which yields
$\langle f, (1-\psi_n)\phi_0\rangle = 0$.
Hence
%
\begin{math}
  \langle f, \psi_n\phi_0\rangle = \langle f, \phi_0\rangle = 1 .
\end{math}
%
However, this is contradicted by
$\| \psi_n\phi_0\|_{ D^{1,p}(\mathbb{R}^N) } \to 0$
as $n\to \infty$, which follows easily from
%
\begin{equation*}
    \| \nabla(\psi_n\phi_0) \|_{ L^p(\mathbb{R}^N) }
  \leq \| \phi_0\|_{ L^\infty(\mathbb{R}^N) }
       \| \nabla\psi_n\|_{ L^p(\mathbb{R}^N) }
    +  \| \nabla\phi_0\|_{ L^\infty(\mathbb{R}^N) }
       \| \psi_n\|_{ L^p(\mathbb{R}^N) }
\end{equation*}
%
with both
%
\begin{gather*}
    \| \nabla\psi_n\|_{ L^p(\mathbb{R}^N) }
  = n^{1 - (N/p)} \| \nabla\psi\|_{ L^p(\mathbb{R}^N) } \to 0,\\
    \| \psi_n\|_{ L^p(\mathbb{R}^N) }
  = n^{- (N/p)} \| \psi\|_{ L^p(\mathbb{R}^N) } \to 0
\end{gather*}
%
as $n\to \infty$, by $1 < p < 2\leq N$.

So let $\phi\in C_{\mathrm{c}}^1(\mathbb{R}^N)$
satisfy condition ($\boldsymbol{\Phi}$).
%
For $\tau\in (0,\infty)$ we compute
%
\begin{equation}
 \int_{\mathbb{R}^N} u_\pm\, \varphi_1^{p-1}\, m \,{\rm d}x
  = \pm\tau
    + \tau^{1 - (p/2)}
    \int_{\mathbb{R}^N} \phi\, \varphi_1^{p-1}\, m \,{\rm d}x ,
\label{u_pm.phi_1}
\end{equation}
%
by
$\int_{\mathbb{R}^N} \varphi_1^p\, m \,{\rm d}x = 1$.
It follows that
%
\[
 \int_{\mathbb{R}^N} u_-\, \varphi_1^{p-1}\, m \,{\rm d}x < 0 <
  \int_{\mathbb{R}^N} u_+\, \varphi_1^{p-1}\, m \,{\rm d}x
  \quad\mbox{for all $\tau > 0$ large enough.}
\]
%
Next we use eqs.\ \eqref{J''.phi_1} and \eqref{def.Q} together with
$\langle f, \varphi_1\rangle = 0$ to obtain
%
\begin{equation}
\begin{split}
&   \mathcal{E}_f(u_\pm)
  = \mathcal{J}_{\lambda_1}
    ( \pm\tau \varphi_1 + \tau^{1 - (p/2)} \phi )
  =
\\
&   \mathcal{Q}_{ \pm \tau^{-p/2} \phi }(\phi,\phi)
  - \tau^{1 - (p/2)}\, \langle f, \phi\rangle
  = \mathcal{Q}_{ \pm \tau^{-p/2} \phi }(\phi,\phi)
  - \tau^{1 - (p/2)} .
\end{split}
\label{E_f:u_pm}
\end{equation}
%
We recall that the quadratic forms
$\mathcal{Q}_{ \pm\tau^{-p/2} \phi }$ are given by
formula \eqref{def.Q}.
Since
$\inf_K |\nabla\varphi_1|$ $> 0$, $\inf_K \varphi_1 > 0$,
and $\phi$ is supported in $K\subset \mathbb{R}^N\setminus \{ 0\}$,
we conclude that both summands in
$\mathcal{Q}_{ \pm\tau^{-p/2} \phi }(\phi,\phi)$
are bounded independently from $\tau\geq \tau_0$,
provided $\tau_0\in (0,\infty)$ is large enough.
Finally, from \eqref{E_f:u_pm} we deduce that
$\mathcal{E}_f(u_\pm) \to -\infty$ as $\tau\to +\infty$.
The conclusion of the lemma follows.
\end{proof}

%
\subsection{A minimax method}
\label{ss:Minimax}

We allow $1<p<N$ throughout the entire paragraph
even though we apply the results to the minimax expression in
\eqref{def.b_lam} only for $p<2$.

We assume that
$0\leq \lambda\leq \Lambda_\infty - \eta$ and
$f\in D^{-1,p'}(\mathbb{R}^N)$.
Here, $\eta$ is an arbitrary, but fixed number with
$0 < \eta < \Lambda_\infty - \lambda_1$.
Furthermore, in view of Lemma~\ref{lem-Ext-Cone}
with $\gamma = \gamma_\eta$,
we find a constant $0 < \gamma_\eta < \infty$ large enough,
so that
$\Lambda_{\gamma_\eta} \geq \Lambda_\infty - \frac12\eta$,
and set
%
\[
 c = \frac12
      \left(
      1 - ( \Lambda_\infty - \eta ) \Lambda_{\gamma_\eta}^{-1}
      \right) > 0 .
\]
%

Note that for any fixed $\tau\in \mathbb{R}$ the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top )$
is coercive on the (closed linear) subspace
$D^{1,p}(\mathbb{R}^N)^\top$ of $D^{1,p}(\mathbb{R}^N)$.
This claim follows from the following inequalities
which are valid whenever
%
\begin{math}
  |\tau|\leq T\leq
  \gamma_\eta^{-1} \| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } ,
\end{math}
%
for any fixed $T\in (0,\infty)$:
%
\begin{equation}
\begin{split}
&   \int_{\mathbb{R}^N} |\nabla ( \tau\varphi_1 + u^\top )|^p \,{\rm d}x
  - (\Lambda_\infty - \eta)
    \int_{\mathbb{R}^N} |( \tau\varphi_1 + u^\top )|^p\, m \,{\rm d}x
\\
& \geq \Big(
       1 - \frac{ \Lambda_\infty - \eta }{ \Lambda_{\gamma_\eta} }
       \Big)
    \int_{\mathbb{R}^N} |\nabla ( \tau\varphi_1 + u^\top )|^p \,{\rm d}x
\\
& \geq \Big(
       1 - \frac{ \Lambda_\infty - \eta }{ \Lambda_{\gamma_\eta} }
       \Big)
    \left|
    \| \nabla u^\top\|_{ L^p(\mathbb{R}^N) }
  - |\tau|\cdot \| \nabla\varphi_1\|_{ L^p(\mathbb{R}^N) }
    \right|^p
\\
& \geq c\, \| \nabla u^\top\|_{ L^p(\mathbb{R}^N) }^p
     - c_T ,
\end{split}
\label{ineq:Lam_1.T}
\end{equation}
%
with another constant $0 < c_T < \infty$ depending solely on $T$.
The first inequality in~\eqref{ineq:Lam_1.T}
is easily derived from formula \eqref{def.Lam_g}.
Consequently, any global minimizer $u_\tau^\top$
for the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top )$
on $D^{1,p}(\mathbb{R}^N)^\top$ satisfies the estimate
$\| u_\tau^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq C_T$ $< \infty$,
where $C_T$ is a constant independent from
$\lambda\in [ 0, \Lambda_\infty - \eta ]$ and $\tau\in [-T,T]$.
Such a global minimizer always exists and verifies
the Euler\--Lagrange equation
%
\begin{equation}
  \begin{split}
&- \Delta_p( \tau\varphi_1 + u_\tau^\top )
  - \lambda\, m(x)\,
    | \tau\varphi_1 + u_\tau^\top |^{p-2}
    ( \tau\varphi_1 + u_\tau^\top ) \\
& = f^\top(x) + \zeta_\tau\cdot m(x)\, \varphi_1(x)^{p-1}
    \quad\mbox{in } \mathbb{R}^N ,
\end{split}
\label{t:BVP.u_tau}
\end{equation}
%
with a Lagrange multiplier $\zeta_\tau\in \mathbb{R}$.
Thus, we may define
%
\begin{equation}
  j_\lambda(\tau)\eqdef
    \min_{ u^\top\in D^{1,p}(\mathbb{R}^N)^\top }
  \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ) .
\label{def.j_lam,tau}
\end{equation}
%
In the rest of our proof of Theorem \ref{thm-Exist:p<2}
in {\S}\ref{ss:Fredh_Mount}
we will show that for $1<p<2$ the function
$j_\lambda\colon \mathbb{R}\to \mathbb{R}$ attains a local maximum
under the conditions of Theorem \ref{thm-Exist:p<2}.

In analogy with the notation
$\mathcal{J}_{\lambda}(u)\equiv \mathcal{J}_{\lambda}(u;f)$,
we write also
$j_\lambda(\tau)\equiv j_\lambda(\tau;f)$
if $f\in D^{-1,p'}(\mathbb{R}^N)$ varies,
to avoid possible confusion.

%
%%%%%    Continuity of Constrained Minimum (Lemma)

\begin{lemma}\label{lem-Cont-Min}
Let\/ $1<p<N$. The mapping
%
\begin{equation}
  (\tau,\lambda,f) \mapsto j_\lambda(\tau;f)\colon
  \mathbb{R}\times [0, \Lambda_\infty - \eta]\times D^{-1,p'}(\mathbb{R}^N) \to \mathbb{R}
\label{j_cont}
\end{equation}
%
is continuous.
In particular, if\/ $0<T<\infty$ and
$K$ is a compact set in $D^{-1,p'}(\mathbb{R}^N)$, then
%
\begin{equation}
  \bigl\{
  j_\lambda(\,\cdot\, ;f)\colon [-T,T]\to \mathbb{R}\colon\
    (\lambda,f)\in [0, \Lambda_\infty - \eta]\times K
  \bigr\}
\label{j_equicont}
\end{equation}
%
is a family of (uniformly) equicontinuous functions.
%
\end{lemma}
%

\begin{proof}
Let $\tau_n\to \tau_0$ in $\mathbb{R}$,
$\mu_n\to \mu_0$ in $[0, \Lambda_\infty - \eta]$, and
$f_n\to f_0$ in $D^{-1,p'}(\mathbb{R}^N)$ as $n\to \infty$.
Suppose that
$j_{\mu_n}(\tau_n;f_n)$ does {\em not\/} converge to
$j_{\mu_0}(\tau_0;f_0)$ as $n\to \infty$.
Passing to a subsequence if necessary, we may assume
%
\begin{equation}
  \liminf_{n\to \infty}
  \left| j_{\mu_n}(\tau_n;f_n) - j_{\mu_0}(\tau_0;f_0) \right|
  > 0 .
\label{j_disc}
\end{equation}
%
Consider any global minimizer $u_n^\top$
for the functional
%
\begin{math}
  u^\top \mapsto
  \mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u^\top ; f_n )
\end{math}
%
on $D^{1,p}(\mathbb{R}^N)^\top$; $n=1,2,\dots$.
The sequence
$\{ u_n^\top \}_{n=1}^\infty$
is bounded in $D^{1,p}(\mathbb{R}^N)$, by ineq.~\eqref{ineq:Lam_1.T},
and hence, it contains a weakly convergent subsequence
(indexed by $n$ again)
$u_n^\top \rightharpoonup w^\top$ in $D^{1,p}(\mathbb{R}^N)^\top$
as $n\to \infty$.
>From the weak lower semicontinuity of
$\mathcal{J}_{\lambda}$ on $D^{1,p}(\mathbb{R}^N)$ we obtain
%
\begin{equation}
\begin{split}
   \liminf_{n\to \infty} j_{\mu_n}(\tau_n;f_n)
&= \liminf_{n\to \infty}
       \mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u_n^\top ; f_n )
\\
& \geq \mathcal{J}_{\mu_0}( \tau_0\varphi_1 + w^\top ; f_0 )
  \geq j_{\mu_0}(\tau_0;f_0) .
\end{split}
\label{j_lower}
\end{equation}
%
On the other hand, if $u_0^\top$ is any global minimizer
for the functional
$u^\top \mapsto$
$\mathcal{J}_{\mu_0}( \tau_0\varphi_1 + u^\top ; f_0 )$
on $D^{1,p}(\mathbb{R}^N)^\top$, then one has
%
\begin{equation}
\begin{split}
   \limsup_{n\to \infty} j_{\mu_n}(\tau_n;f_n)
& \leq \lim_{n\to \infty}
    \mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u_0^\top ; f_n )
\\
& = \mathcal{J}_{\mu_0}( \tau_0\varphi_1 + u_0^\top ; f_0 )
  = j_{\mu_0}(\tau_0;f_0) .
\end{split}
\label{j_upper}
\end{equation}
%
We combine inequalities \eqref{j_lower} and~\eqref{j_upper}
to get
\[
  \lim_{n\to \infty} j_{\mu_n}(\tau_n;f_n) = j_{\mu_0}(\tau_0;f_0)
\]
which contradicts~\eqref{j_disc}.
The continuity of
$(\tau,\lambda,f) \mapsto j_\lambda(\tau;f)$ is proved.

Finally, the equicontinuity of the family~\eqref{j_equicont}
is a consequence of the uniform continuity of
the mapping~\eqref{j_cont} on the compact set
$[-T,T]\times [0, \Lambda_\infty - \eta]\times K$.
\end{proof}


%%%%%    Continuity of Constrained Minimum (Remark)

\begin{remark}\label{rem-Cont-Min}
\begingroup\rm
We claim that in the proof of Lemma~\ref{lem-Cont-Min},
$w^\top$ is a global minimizer for the functional
%
\begin{math}
  u^\top \mapsto
  \mathcal{J}_{\mu_0}( \tau_0\varphi_1 + u^\top ; f_0 )
\end{math}
%
on $D^{1,p}(\mathbb{R}^N)^\top$ and we have also
$u_n^\top \to w^\top$ strongly in $D^{1,p}(\mathbb{R}^N)$
as $n\to \infty$.
First of all,
\eqref{j_lower} and~\eqref{j_upper} imply
\[
      j_{\mu_n}(\tau_n;f_n)
    = \mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u_n^\top ; f_n )
  \to \mathcal{J}_{\mu_0}( \tau_0\varphi_1 +   w^\top ; f_0 )
    = j_{\mu_0}(\tau_0;f_0) .
\]
Combining this result with
$\tau_n\to \tau_0$, $\mu_n\to \mu_0$,
$f_n\to f_0$ in $D^{-1,p'}(\mathbb{R}^N)$,
$u_n^\top \rightharpoonup w^\top$ weakly in $D^{1,p}(\mathbb{R}^N)$,
and
$u_n^\top \to w^\top$ strongly in $L^p(\mathbb{R}^N; m)$, we arrive at
\[
  \left\| \tau_n\varphi_1 + u_n^\top \right\|_{ D^{1,p}(\mathbb{R}^N) }
    \to
  \left\| \tau_0\varphi_1 +   w^\top \right\|_{ D^{1,p}(\mathbb{R}^N) }
    \quad\mbox{as } n\to \infty .
\]
Thus, the uniform convexity of the standard norm on
$D^{1,p}(\mathbb{R}^N)$ forces
$\tau_n\varphi_1 + u_n^\top \to \tau_0\varphi_1 + w^\top$
strongly in $D^{1,p}(\mathbb{R}^N)$.
Our claim now follows as $\tau_n\to \tau_0$.
\endgroup
\end{remark}
%

Obviously, if the function
$j_{\lambda}\colon \mathbb{R}\to \mathbb{R}$ has a local minimum
at some point $\tau_0\in \mathbb{R}$, and
$u_0^\top$ is a global minimizer for the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u^\top )$
on $D^{1,p}(\mathbb{R}^N)^\top$, then
$u_0 = \tau_0\varphi_1 + u_0^\top$ is a local minimizer for
$\mathcal{J}_{\lambda}$ on $D^{1,p}(\mathbb{R}^N)$ and thus
a weak solution to problem~\eqref{e:BVP.l}.
Our next lemma displays a similar result if
$j_{\lambda}$ has a {\it local maximum\/} at $\tau_0\in \mathbb{R}$;
it claims that
$\beta_{\lambda}$ in~\eqref{def.b_lam}
is a critical value of $\mathcal{J}_{\lambda}$.

%%%%%    Criticality of the Maximin Point (Lemma)

\begin{lemma}\label{lem-Crit-Value}
Let\/
$0\leq \lambda\leq \Lambda_\infty - \eta$ and\/
$f\in D^{-1,p'}(\mathbb{R}^N)$.
Assume that the function
$j_{\lambda}\colon \mathbb{R}\to \mathbb{R}$ attains a local maximum
$\beta_{\lambda}$ at some point $\tau_0\in \mathbb{R}$.
Then there exists
$u_0^\top\in D^{1,p}(\mathbb{R}^N)^\top$ such that\/
$u_0^\top$ is a global minimizer for the functional\/
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u^\top )$
on $D^{1,p}(\mathbb{R}^N)^\top$,
$u_0 = \tau_0\varphi_1 + u_0^\top$ is a critical point for
$\mathcal{J}_{\lambda}$, and
$\mathcal{J}_{\lambda}(u_0) = \beta_{\lambda}$.
%
\end{lemma}

\begin{proof}
Given an arbitrary numerical sequence
$\{ \tau_n\}_{n=1}^\infty$ with
$\tau_n\to \tau_0$ in~$\mathbb{R}$ as $n\to \infty$ and
$\tau_n\not= \tau_0$ for all $n\geq 1$,
we can deduce from Remark~\ref{rem-Cont-Min}
that this sequence contains a subsequence
denoted again by $\{ \tau_n\}_{n=1}^\infty$, such that
for each $n=0,1,2,\dots$,
$u_n^\top$ is a global minimizer for the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau_n\varphi_1 + u^\top )$
and
$u_n^\top \to u_0^\top$ strongly in $D^{1,p}(\mathbb{R}^N)$
as $n\to \infty$.
It follows that
%
\begin{equation}
\begin{split}
  \mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
  - \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_n^\top )
& \leq
   \mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
  - \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_0^\top )\\
&= j_{\lambda}(\tau_n) - j_{\lambda}(\tau_0)
  \leq 0
\end{split}
\label{c:J_upper}
\end{equation}
%
for all integers $n\geq 1$ sufficiently large;
again, we may assume it for all $n\geq 1$.

On the other hand, denoting
\[
  \phi_n(s)\eqdef
  \tau_0\varphi_1 + u_n^\top + s(\tau_n - \tau_0) \varphi_1
    \quad\mbox{for }\ 0\leq s\leq 1 ;\ n\geq 1 ,
\]
we have
%
\begin{eqnarray*}
&   \mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
  - \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_n^\top )
  = (\tau_n - \tau_0) \int_0^1
    \left\langle
      \mathcal{J}_{\lambda}^\prime (\phi_n(s)) ,\, \varphi_1
    \right\rangle \,{\rm d}s
\end{eqnarray*}
%
where
%
\begin{align*}
    \left\langle
      \mathcal{J}_{\lambda}^\prime (\phi_n(s)) ,\, \varphi_1
    \right\rangle
& = \int_{\mathbb{R}^N}
      |\nabla \phi_n(s)|^{p-2}\, \nabla \phi_n(s)
        \cdot \nabla\varphi_1 \,{\rm d}x\\
& \quad
  - \lambda \int_{\mathbb{R}^N}
      |\phi_n(s)|^{p-2}\, \phi_n(s)\, \varphi_1\, m \,{\rm d}x
  - \int_{\mathbb{R}^N} f\varphi_1 \,{\rm d}x .
\end{align*}
%
Since
$\phi_n(s)\to u_0 = \tau_0\varphi_1 + u_0^\top$
strongly in $D^{1,p}(\mathbb{R}^N)$ and
uniformly for $0\leq s\leq 1$, we arrive at
%
\begin{equation}
\begin{split}
&   (\tau_n - \tau_0)^{-1}
      \left[
    \mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
  - \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_n^\top )
      \right]
\\
& \longrightarrow\;
    \left\langle
      \mathcal{J}_{\lambda}^\prime (u_0) ,\, \varphi_1
    \right\rangle
  = \zeta_0\, \| \varphi_1\|_{ L^p(\mathbb{R}^N; m) } = \zeta_0
    \quad\mbox{as } n\to \infty ,
\end{split}
\label{c:J_lower}
\end{equation}
%
where
%
\begin{align*}
     \left\langle
      \mathcal{J}_{\lambda}^\prime (u_0) ,\, \varphi_1
    \right\rangle
& = \int_{\mathbb{R}^N}
      |\nabla u_0|^{p-2}\, \nabla u_0
        \cdot \nabla\varphi_1 \,{\rm d}x\\
& \quad
  - \lambda \int_{\mathbb{R}^N} |u_0|^{p-2} u_0\varphi_1\, m \,{\rm d}x
  - \int_{\mathbb{R}^N} f\varphi_1 \,{\rm d}x
\end{align*}
and
$\zeta_0\in \mathbb{R}$ is the Lagrange multiplier given by
$\mathcal{J}_{\lambda}^\prime (u_0) = \zeta_0\, m\, \varphi_1^{p-1}$.

Finally, if we choose $\tau_n$ such that
the sign of $(\tau_n - \tau_0)$
does not change for all $n=1,2,\dots$, then
\eqref{c:J_upper} and~\eqref{c:J_lower} yield
$\zeta_0\leq 0$ if $\mathop{\rm sgn}(\tau_n - \tau_0) = 1$, and
$\zeta_0\geq 0$ if $\mathop{\rm sgn}(\tau_n - \tau_0) = -1$.
Since both alternatives are possible, we conclude that
$\zeta_0 = 0$ which shows
$\mathcal{J}_{\lambda}^\prime (u_0) = 0$, i.e.,
$u_0$ is a weak solution to problem~\eqref{e:BVP.l_1} as desired.
In particular,
$\mathcal{J}_{\lambda}(u_0)$
is a critical value of $\mathcal{J}_{\lambda}$.
\end{proof}


%%%%%    Criticality of the Maximin Point (Remark)

\begin{remark}\label{rem-Crit-Value}
\begingroup\rm
As an easy consequence of
\eqref{c:J_upper}, \eqref{c:J_lower}
in the proof of Lemma~\ref{lem-Crit-Value},
we conclude that the function
$j_{\lambda}\colon \mathbb{R}\to \mathbb{R}$ is differentiable at $\tau_0$ with
$j_{\lambda}^\prime (\tau_0) = 0$.
\endgroup
\end{remark}

%
\subsection{Rest of the proof of Theorem \ref{thm-Exist:p<2}}
\label{ss:Fredh_Mount}

We deduce from Lemma~\ref{lem-Saddle_Geom} that there exist
$a,b\in \mathbb{R}$ such that $a<0<b$ and
%
\[
 \max\{ j_{\lambda_1}(a;f^{\#}) ,\, j_{\lambda_1}(b;f^{\#}) \}
    < j_{\lambda_1}(0;f^{\#}) .
\]
%
The ``continuity'' Lemma~\ref{lem-Cont-Min} shows that
there exist numbers
$\delta\equiv \delta(f^{\#}) > 0$ and
$\varrho\equiv \varrho(f^{\#}) > 0$ such that also with
$f = f^{\#} + \zeta\, m\varphi_1^{p-1}$
we have
%
\begin{eqnarray*}
& \max\{ j_{\lambda}(a;f) ,\, j_{\lambda}(b;f) \}
    < j_{\lambda}(0;f)
\end{eqnarray*}
%
for all
$\lambda\in (\lambda_1 - \delta, \lambda_1 + \delta)$
and all
$\zeta\in (-\varrho,\varrho)$.
Now we can apply Lemma~\ref{lem-Crit-Value}
to conclude that the functional
$\mathcal{J}_{\lambda}(\,\cdot\, ;f)$
possesses a critical point
$u_1 = \tau_1\varphi_1 + u_1^\top$, with some
$\tau_1\in (a,b)$ and
$u_1^\top\in D^{1,p}(\mathbb{R}^N)^\top$.
This proves Theorem \ref{thm-Exist:p<2}.


\begin{proof}[Proof of Remark~\ref{rem-Exist:p<2}]
If $\lambda < \lambda_1$ then we have
$j_{\lambda}(\tau;f) \to +\infty$ as $|\tau|\to \infty$.
Consequently, for
$\lambda\in (\lambda_1 - \delta, \lambda_1)$
and
$\zeta\in (-\varrho,\varrho)$,
the continuous function
$j_{\lambda}(\,\cdot\, ;f) \colon \mathbb{R}\to \mathbb{R}$
possesses also a local minimizer in each of the intervals
$(-\infty, \tau_1)$ and $(\tau_1, \infty)$, say,
$\tau_2$ and $\tau_3$, respectively.
Our definition of $j_{\lambda}(\,\cdot\, ;f)$ now shows that
$u_2 = \tau_2\varphi_1 + u_2^\top$ and
$u_3 = \tau_3\varphi_1 + u_3^\top$
are local minimizers for
$\mathcal{J}_{\lambda}(\,\cdot\, ;f)$, with some
$u_2^\top, u_3^\top\in D^{1,p}(\mathbb{R}^N)^\top$, as claimed.
\end{proof}


%%%%%    APPENDIX: ASYMPTOTICS OF THE FIRST EIGENFUNCTION

\section{Appendix: Asymptotics of the eigenfunction $\varphi_1$}
\label{s:Asymptotic}

To determine the asymptotic behavior of
the first eigenfunction $\varphi_1$ of
the $p$-Laplacian $\Delta_p$ on $\mathbb{R}^N$
subject to a weight $m(|x|)$, for $1<p<N$,
we consider a strictly positive, radially symmetric function
$u\colon \mathbb{R}^N\to (0,\infty)$ of class $C^1$,
$u(x)\equiv u(r)$ with $r\equiv |x|$, $x\in \mathbb{R}^N$,
which satisfies the following partial differential equation
(in the sense of distributions on $\mathbb{R}^N$):
%
\begin{equation}
\label{ev:u}
  - \Delta_p u = m(|x|)\, u^{p-1} \quad \mbox{ for } x\in \mathbb{R}^N ;
  \quad
  u(|x|)\to 0 \;\mbox{ as } |x|\to \infty .
\end{equation}
%
We weaken the strict positivity in
hypothesis \eqref{hyp:m>0} on the weight $m$ as follows:

%%%%%%%%%%     MAIN HYPOTHESIS (weaker)     %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
\renewcommand{\labelenumi}{({\bf H'})}
%
\item
\makeatletter
\def\@currentlabel{{H'}} \label{hyp:m=0}
\makeatother
There exist constants $\delta > 0$ and $C>0$ such that
%
\begin{equation}
  0\leq m(r)\leq \frac{C}{ (1+r)^{p+\delta} }
  \quad\mbox{for almost all }\ 0\leq r < \infty ,
\label{ineq:m=0}
\end{equation}
%
and $m\not\equiv 0$ in $\mathbb{R}_+$.
\end{enumerate}
%
Under this hypothesis, we are able to establish
the following asymptotic behavior of
$u(r)$ and $u'(r)$ as $r\to \infty$.

%
%%%%%    PROPOSITION: ASYMPTOTICS OF THE FIRST EIGENFUNCTION    %%%%%%%
%
\begin{proposition}\label{prop-Asympt}
There exists a constant $c > 0$ such that
%
\begin{gather}
\lim_{r\to \infty}
  \left( u(r)\, r^{ \frac{N-p}{p-1} } \right) = c ,
\label{e:u.infty} \\
\lim_{r\to \infty}
  \left( u'(r)\, r^{ \frac{N-1}{p-1} } \right)
  =  - \textstyle\frac{N-p}{p-1}\, c .
\label{e:u'.infty}
\end{gather}
%
\end{proposition}
%

For the related Cauchy problem,
%
\begin{equation}
\label{ev:u:rel}
  - \Delta_p u(|x|) = f(u(|x|)) \;\mbox{ for } x\in \mathbb{R}^N ;
  \qquad
  u(|x|)\to 0 \;\mbox{ as } |x|\to \infty ,
\end{equation}
%
with $f(u)\geq 0$ for $u>0$ sufficiently small,
the inequalities
$$
  u(r)\, r^{ \frac{N-p}{p-1} } \geq c_1 > 0
    \quad\mbox{and }\quad
  - u'(r)\, r^{ \frac{N-1}{p-1} } \geq c_2 > 0
$$
for all sufficiently large $r>0$
(with some constants $c_1$ and $c_2$)
have been established in the work of
{\sc Ni} and {\sc Serrin} \cite[Theorem 6.1]{NiSerrin}.
Their method of proof applies also to our case.
%
For the inequality
%
\begin{equation}
\label{ev:u:ineq}
  - \Delta_p u\leq m(|x|)\, u^{p-1} \quad\mbox{for } x\in \mathbb{R}^N ;
  \quad
  u(x)\to 0 \;\mbox{ as } |x|\to \infty ,
\end{equation}
%
with $u(x)$ not necessarily radially symmetric, $u(x) > 0$,
but with the weight $m(r)$ decaying at infinity faster than ours,
an upper estimate on the decay of $u$ at infinity can be found in
{\sc Fleckinger}, {\sc Harrell} and {\sc de Th\'elin}
\cite[Theorem IV.2]{FHdT}.

In the proof of Proposition~\ref{prop-Asympt}
we need a few auxiliary results.

The Cauchy problem \eqref{ev:u} is equivalent to
%
\begin{equation}
\label{ev:u.rad}
\begin{gathered}
{}- (|u'|^{p-2} u')' - \frac{N-1}{r}\, |u'|^{p-2} u'
  = m(r)\, u^{p-1} \quad\mbox{for } r>0 ;
\\
  u'(r)\to 0 \mbox{ as } r\to 0 \quad\mbox{and }\quad
  u(r) \to 0 \mbox{ as } r\to \infty .
\end{gathered}
\end{equation}
%
This problem can be rewritten as
%
\begin{equation}
\label{ev:r.u.rad}
\begin{gathered}
{}- ( r^{N-1}\, |u'|^{p-2} u' )' = m(r)\, r^{N-1}\, u^{p-1}
    \quad\mbox{for } r>0 ; \\
  u'(r)\to 0 \mbox{ as } r\to 0 \quad\mbox{and }\quad
  u(r) \to 0 \mbox{ as } r\to \infty .
\end{gathered}
\end{equation}
%
We reduce this second-order differential equation to
a first-order equation by introducing
the Riccati-type transformation
%
\begin{equation}
\label{u:Riccati}
  U(r)\eqdef
{}- r^{p-1}
  \Big| \frac{ u'(r) }{ u(r) } \Big|^{p-2}
         \frac{ u'(r) }{ u(r) }
  \quad\mbox{for } r>0 ,\quad
  U(0)\eqdef 0 .
\end{equation}
%
By~\eqref{ev:r.u.rad}, the function
$r\mapsto r^{ \frac{N-1}{p-1} } u'(r)$
is nonincreasing for $0 < r < \infty$ which implies
$u'(r)\leq 0$ for all $r>0$, and therefore also
$U(r)\geq 0$.
Hence, for $r>0$,
%
\begin{align*}
  U'(r)
& =
  - (p-1) r^{p-2}
  \big| \frac{u'}{u} \big|^{p-2} \frac{u'}{u}
  - (p-1) r^{p-1}
  \big| \frac{u'}{u} \big|^{p-2}
    \big[ \frac{u''}{u} - \big(\frac{u'}{u}\big)^2 \big]
\\
& = \frac{p-1}{r}\, U(r)
  - r^{p-1}\, \frac{ ( |u'|^{p-2} u' )' }{ |u|^{p-2} u }
  + \frac{p-1}{r}\, U(r)^{ \frac{p}{p-1} } .
\end{align*}
%
Inserting the second derivative expression from
equation \eqref{ev:u.rad}, we arrive at
\[
  U'(r) =
  - \frac{N-p}{r}\, U(r)
  + \frac{p-1}{r}\, U(r)^{ \frac{p}{p-1} }
  + m(r)\, r^{p-1} .
\]
This is a differential equation for the unknown function $U$
which we rewrite as
%
\begin{equation}
\label{eq:U}
  U'(r)
  = \frac{p-1}{r}\, U(r)
    \Big( U(r)^{ \frac{1}{p-1} } - \frac{N-p}{p-1} \Big)
  + m(r)\, r^{p-1}
    \quad\mbox{for } r>0 .
\end{equation}
%

An upper bound for $U(r)$ is obtained first:

%%%%%    LEMMA - Upper Bound for U(r)

\begin{lemma}\label{lem-UpperBound}
We have
%
\begin{equation}
\label{e:U.Upper}
  U(r)\leq c_{N,p}\eqdef \big(\frac{N-p}{p-1}\big)^{p-1}
  \quad\mbox{for all } r\geq 0 .
\end{equation}
\end{lemma}


\begin{proof}
Clearly, by \eqref{u:Riccati}, the function
$U\colon \mathbb{R}_+\to \mathbb{R}$ is continuous and, by \eqref{eq:U},
it is differentiable almost everywhere with the derivative
$U'$ being locally bounded.
Now, in contradiction with \eqref{e:U.Upper},
suppose that there exists a number $r_0\geq 0$ such that
$U(r_0) > c_{N,p}$.
Let
$$
  r_1\eqdef
  \sup\{ r'\colon r'\geq r_0 \,\mbox{ and }\, U(r) > c_{N,p}
         \;\mbox{ for all }\; r_0\leq r\leq r' \} .
$$
Next we show that $r_1 = \infty$.
Indeed, equation \eqref{eq:U} with
$U(r) > c_{N,p}$ and $m(r)\geq 0$ for $r_0\leq r < r_1$ implies
$U'(r) > 0$.
This shows that the function $U(r)$ is strictly increasing for
$r_0\leq r < r_1$.
Consequently, $r_1 < \infty$ would yield
$U(r_1) = c_{N,p} < U(r_0) < U(r_1)$ which is impossible.

Hence, there is a constant $\gamma > 0$ such that
the expression inside the parenthesis in eq.~\eqref{eq:U}
satisfies
%
\[
  U(r)^{ \frac{1}{p-1} } - \frac{N-p}{p-1}
  \geq \gamma\, U(r)^{ \frac{1}{p-1} }
    \quad\mbox{for all } r\geq r_0 .
\]
%
Applying this inequality to equation \eqref{eq:U}
we obtain
%
\[
 U'(r)\geq \frac{p-1}{r}\, \gamma\, U(r)^{ \frac{p}{p-1} }
    \quad\mbox{for all } r\geq r_0 .
\]
%
We integrate this inequality over the interval
$[r_0,r]$ to get
$$
  U(r_0)^{ - \frac{1}{p-1} } - U(r)^{ - \frac{1}{p-1} }
  \geq \gamma\, \log (r/r_0)
    \quad\mbox{for all } r\geq r_0 .
$$
Recalling $U(r) > 0$ and letting $r\to \infty$,
we arrive at
$U(r_0)^{ - \frac{1}{p-1} } \geq +\infty$,
which is a contradiction.
Inequality \eqref{e:U.Upper} is proved.
\end{proof}

Define the function
%
\begin{equation}
\label{def:a(r)}
  a(r)\eqdef
    \frac{p-1}{r}
    \Big( \frac{N-p}{p-1} - U(r)^{ \frac{1}{p-1} } \Big)
  \quad\mbox{for } r>0 .
\end{equation}
%
Note that $a(r)\geq 0$ by Lemma~\ref{lem-UpperBound},
and
$$
  a(r) = \frac{N-p}{r} + (p-1) \,\frac{u'(r)}{u(r)}
       = (p-1) \,\frac{ {\rm d} }{ {\rm d}r }
  \log\left( u(r)\, r^{ \frac{N-p}{p-1} } \right) .
$$
We substitute this function into eq.~\eqref{eq:U}
and use integrating factor to integrate it over
any interval $[r_0,r]$ with $r_0 > 0$ fixed and $r\geq r_0$.
We thus obtain
%
\begin{equation}
\label{eq:U(r)}
    U(r) - U(r_0)\, e^{ - \int_{r_0}^r a(s) \,{\rm d}s }
  = \int_{r_0}^r m(s)\, s^{p-1}\,
    e^{ - \int_s^r a(t) \,{\rm d}t } \,{\rm d}s .
\end{equation}
%
Furthermore, we introduce the abbreviation
%
\begin{equation}
\label{def:A(r)}
    A(r)\eqdef \int_{r_0}^r a(s) \,{\rm d}s
  = (p-1)\, \log
    \frac{ u(r)\,   r^{ \frac{N-p}{p-1} } }
         { u(r_0)\, r_0^{ \frac{N-p}{p-1} } }
  \quad\mbox{for } r\geq r_0 .
\end{equation}
%

%%%%%    LEMMA - Lower Bound for U(r)

\begin{lemma}\label{lem-LowerBound}
We have $a(r)\geq 0$ for all $r>0$ and
%
\begin{equation}
\label{e:U.Lower}
  \int_{r_0}^\infty a(r) \,{\rm d}r < \infty
  \quad\mbox{for every } r_0 > 0 .
\end{equation}
%
\end{lemma}
%

\begin{proof}
The function $A(r)$ is nondecreasing for $r_0\leq r < \infty$.
Now, suppose that
$\lim_{r\to \infty} A(r) = +\infty$.
 From equation \eqref{eq:U(r)} we deduce
%
\begin{equation}
\label{ineq:U(r)}
  0 \leq U(r) - U(r_0)\, e^{- A(r)} \leq
    \int_{r_0}^\infty m(s)\, s^{p-1}\,
    e^{ - ( A(r) - A(s) ) } \,{\rm d}s
  \quad\mbox{for } r\geq r_0 .
\end{equation}
%
Due to our hypothesis \eqref{hyp:m=0}, we are allowed to apply
Lebesgue's dominated convergence theorem to the last integral
to obtain, as $r\to \infty$,
$0\leq \lim_{r\to \infty} U(r)\leq 0$, i.e.,
$\lim_{r\to \infty} U(r) = 0$.

This shows that, given any number $\eta$ such that
$0 < \eta < N-p$,
there exists a number $r_\eta\geq r_0$ such that
$$
  a(r) =
    \frac{p-1}{r}
    \Big( \frac{N-p}{p-1} - U(r)^{ \frac{1}{p-1} } \Big)
  \geq \frac{N - p - \eta}{r}
  \quad\mbox{for all } r\geq r_\eta .
$$
Since $r_0$ is arbitrary, $r_0 > 0$,
we may take $r_0 = r_\eta$.
Upon integration, we get
%
\begin{equation}
\label{ineq:A(r)}
  A(r)\geq (N - p - \eta) \int_{r_0}^r \frac{ {\rm d}s }{s}
  = \log \left( (r/r_0)^{N - p - \eta} \right)
  \quad\mbox{for all } r\geq r_0 .
\end{equation}
%
We apply inequalities
\eqref{ineq:m=0} and \eqref{ineq:A(r)} to equation \eqref{eq:U(r)}
to obtain for all $r\geq 0$,
%
\begin{equation}
\label{est:|U|}
\begin{split}
  U(r)&\leq
    U(r_0) \big(\frac{r}{r_0}\big)^{ - (N - p - \eta) }
  + C \int_{r_0}^r \frac{ s^{p-1} }{ (1+s)^{p+\delta} }
             \big(\frac{r}{s}\big)^{ - (N - p - \eta) }
      \,{\rm d}s \\
& \leq
    U(r_0) \big(\frac{r}{r_0}\big)^{ - (N - p - \eta) }
  + \frac{ C\, r^{ - (N - p - \eta) } }{ N - p - \eta - \delta }
    \left(
        r^{   N - p - \eta - \delta }
      - r_0^{ N - p - \eta - \delta }
    \right) .
\end{split}
\end{equation}
%
Note that in inequality~\eqref{ineq:m=0},
the constant $\delta > 0$ may be chosen arbitrarily small;
we choose it such that
$0 < \delta < N - p - \eta$.
Hence, \eqref{est:|U|} yields
$$
  U(r)\leq C_0\, r^{-\delta}
  \quad\mbox{for all } r\geq r_0 ,
$$
where $C_0 > 0$ is a constant.
With our definition of $U$ we have equivalently
$$
- \frac{u'(r)}{u(r)} \leq
  C_0^{ \frac{1}{p-1} }\, r^{ - 1 - \frac{\delta}{p-1} }
  \quad\mbox{for all } r\geq r_0 .
$$
Upon integration we get
$$
{}- \log \frac{u(r)}{u(r_0)}
  \leq C_0'
    \big(
      r_0^{ - \frac{\delta}{p-1} }
      - r^{ - \frac{\delta}{p-1} }
    \big)
  \quad\mbox{for all } r\geq r_0 ,
$$
where $C_0'> 0$ is a constant.
Recalling $u(r)\to 0$ as $r\to \infty$,
we arrive at
$+\infty\leq C_0'\, r_0^{ - \frac{\delta}{p-1} }$
which is absurd.
The proof of the lemma is complete.
\end{proof}


Finally, we determine the limit of the function $U$ at infinity.

%%%%%    LEMMA - Limit of U(r)

\begin{lemma}\label{lem-U.Limit}
We have
%
\begin{equation}
\label{e:U.Limit}
    \lim_{r\to \infty} U(r) = c_{N,p}
  = \big(\frac{N-p}{p-1}\big)^{p-1} .
\end{equation}
\end{lemma}
%

\begin{proof}
The limit
\[
 A(\infty)\eqdef \lim_{r\to \infty} A(r)
  = \int_{r_0}^\infty a(r) \,{\rm d}r
\]
exists and satisfies $0\leq A(\infty) < \infty$,
by \eqref{def:A(r)} and~\eqref{e:U.Lower}.
We apply this fact and hypothesis \eqref{hyp:m=0}
to equation \eqref{eq:U(r)} to obtain the existence of the limit
%
\begin{equation}
\label{lim:U(r)}
  U(\infty)\eqdef \lim_{r\to \infty} U(r)
  = U(r_0)\, e^{- A(\infty)}
  + \int_{r_0}^\infty m(s)\, s^{p-1}\,
    e^{ - ( A(\infty) - A(s) ) } \,{\rm d}s ,
\end{equation}
%
using Lebesgue's dominated convergence theorem.
We have $U(\infty)\leq c_{N,p}$ by~\eqref{e:U.Upper}.
However, if $U(\infty) < c_{N,p}$ then
there exist constants $\gamma > 0$ and $r_1\geq r_0$ such that
$$
  a(r) =
    \frac{p-1}{r}
    \Big( \frac{N-p}{p-1} - U(r)^{ \frac{1}{p-1} } \Big)
  \geq \frac{\gamma}{r}
  \quad\mbox{for all } r\geq r_1 .
$$
But this inequality contradicts~\eqref{e:U.Lower}.
We have proved~\eqref{e:U.Limit}.
\end{proof}


Finally, we are ready to derive formulas
\eqref{e:u.infty} and~\eqref{e:u'.infty}.


\begin{proof}[Proof of Proposition~\ref{prop-Asympt}]
We combine \eqref{def:A(r)} and \eqref{e:U.Lower}
to conclude that the limit
$$
  c_0\eqdef \lim_{r\to \infty} \log
  \Big(
         { u(r)\,   r^{ \frac{N-p}{p-1} } } \Big\slash
         { u(r_0)\, r_0^{ \frac{N-p}{p-1} } }
  \Big)
$$
exists and satisfies $0\leq c_0 < \infty$.
The desired formula \eqref{e:u.infty}
follows immediately with
$c\eqdef e^{c_0}\, u(r_0)\, r_0^{ \frac{N-p}{p-1} } > 0$.
%
The convergence formula \eqref{e:U.Limit} reads
%
\begin{equation}
\label{e:u'/u.infty}
{}- r \,\frac{u'(r)}{u(r)} \,\longrightarrow\,
    \frac{N-p}{p-1}
  \quad\mbox{as } r\to \infty .
\end{equation}
%
We combine this result with \eqref{e:u.infty}
to get~\eqref{e:u'.infty}.
%
The proposition is proved.
\end{proof}

\subsection*{Acknowledgment}
This work was supported in part by
le Minist\`ere des Affaires \'Etrang\`eres (France)
and
the German Academic Exchange Service (DAAD, Germany)
within the exchange program ``PROCOPE''.

 A part of this research was performed when
 P.~T.\ was a visiting professor at CEREMATH,
 Universit\'e Toulouse~1 -- Sciences Sociales,
  Toulouse, France.

A part of the research reported here was performed when
Peter Tak\'a\v{c} was a visiting professor at CEREMATH,
Universit\'e Toulouse~1 -- Sciences Sociales, Toulouse, France.

The authors express their thanks to an anonymous referee for
careful reading of the manuscript and a couple of pertinent references.

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\end{thebibliography}

\end{document}