\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 112, pp. 1--8.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/112\hfil A resonance problem]
{A resonance problem for the p-laplacian in $\mathbb{R}^N$}
\author[G. Izquierdo B., G. L\'{o}pez G.\hfil EJDE-2005/112\hfilneg]
{Gustavo Izquierdo Buenrostro  \& Gabriel L\'{o}pez Garza}

\address{Gustavo Izquierdo Buenrostro \hfil\break
Dept. Mat. Universidad Aut\'{o}noma Metropolitana\\
M\'{e}xico}
\email{iubg@xanum.uam.mx}

\address{Gabriel L\'{o}pez Garza \hfill\break
Dept. Mat. Universidad Aut\'{o}noma Metropolitana\\
M\'{e}xico}
\email{grlzgz@xanum.uam.mx}

\date{}
\thanks{Submitted May 31, 2005. Published October 17, 2005.}
\subjclass[2000]{35J20}
\keywords{Resonance; $p$-Laplacian; improved Poincar\'{e} inequality}

\begin{abstract}
 We show the existence of a weak solution for the problem
 $$
 -\Delta_p u=\lambda_1h(x)|u|^{p-2}u+a(x)g(u)+f(x),\quad
 u\in\mathcal{D}^{1,p}(\mathbb{R}^N),
 $$
 where, $2<p<N$, $\lambda_1$ is the first eigenvalue of the
 $p$-Laplacian on $\mathcal{D}^{1,p}(\mathbb{R}^N)$ relative to
 the radially symmetric weight $h(x)=h(|x|)$. In this problem,
 $g(s)$ is a bounded function for all $s\in\mathbb{R}$,
 $a\in L^{(p^{*})'}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$ and
 $f\in L^{(p^{*})'}(\mathbb{R}^N)$. To establish an existence result,
 we employ the Saddle Point Theorem of Rabinowitz \cite{R} and an
 improved Poincar\'{e} inequality from an  article of Alziary,
 Fleckinger and Tak\'{a}\u{c} \cite{Takac}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lema}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}
Resonance problems for divergence operators have been of interest
since the 197O's. For the ordinary Laplacian on bounded domains
there are a number of classical papers and some recent papers
explore resonant problems in  $\mathbb{R}^N$. For the $p$-Laplacian,
a family of resonant problems in $\mathbb{R}^N$ has been studied
just recently in \cite{Takac} among others. In this paper, we
study the family of $p$-Laplacian equations:
\begin{equation}\label{eq1}
-\Delta_p u=\lambda h(x)|u|^{p-2}u+a(x)g(u)+f(x)\quad
\mbox{in }\mathcal{D}^{1,p}(\mathbb{R}^N),
\end{equation}
where $\Delta_p u= \mathop{\rm div} (|\nabla u|^{p-2}\nabla u)$,
$2<p<N$, ($N\geqslant 3$); $f\in L^{(p^{*})'}(\mathbb{R}^N)$,
$p^{*}=\frac{Np}{N-p}$ and $(p^*)'$  denotes the conjugate of
$p^*$; $g:\mathbb{R}\rightarrow\mathbb{R}$ is a bounded continuous
function ($|g(s)|\leqslant M$) for all $s\in \mathbb{R}$; the
function $h\in L^{N/p}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$,
$h\geqslant 0$ a.e. is a weight function and
$a\in L^{(p^{*})'}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$.
 As usual, the space $\mathcal{D}^{1,p}(\mathbb{R}^N)$ is the closure of
$\mathcal{C}_0^{\infty}(\mathbb{R}^N)$ with respect to the norm
$$
\|u\|=\Big( \int |\nabla u|^p\Big)^{1/p}.
$$
>From here and henceforth the integrals and all the spaces are
taken over
 $\mathbb{R}^N$  unless otherwise specified.

The term resonance is well known in the literature, and refers to the case
in which $\lambda$ is an eigenvalue of the problem
\begin{equation}\label{eigen}
\begin{gathered}
-\Delta_p u=\lambda h(x)|u|^{p-2}u,\\
u\in \mathcal{D}^{1,p}.
\end{gathered}
\end{equation}
 In  \cite{All}, Allegreto et al. show that the eigenvalue problem
\eqref{eigen} possesses a sequence of eigenvalues
 $0<\lambda_1<\lambda_2\leqslant,\dots$ and a corresponding sequence
 of eigenfunctions $\{\varphi_j\}$, where $\varphi_1$ can be chosen to be
positive a.e.. Moreover, we have the Rayleigh quotient characterization:
\begin{equation}
\label{Ray}
\lambda_1=\inf\Big\{ \int |\nabla u|^p: u\in \mathcal{D}^{1,p}\mbox{ with }
\int h |u|^p=1  \Big\}.
 \end{equation}
 We consider the function $\varphi_1$ to be normalized; i.e.,
  $\int h|\varphi_1|^p=1$ and we decompose any function $u\in\mathcal{D}^{1,p}$
as a direct sum
 \begin{equation}
 \label{sum}
 \begin{gathered}
 u=\alpha \varphi_1+w\mbox{ where }\\
 \alpha=\int h|\varphi_1|^{p-2}\varphi_1u\mbox{ and }\int h|\varphi_1|^{p-2}\varphi_1w=0.
 \end{gathered}
 \end{equation}
 Hence, we introduce the spaces
 \begin{equation}
 \label{spaces}
 \begin{gathered}
 V\stackrel{\text{def}}{=}\mathop{\rm span}\{\varphi_1\},\\
 W \stackrel{\text{def}}{=}\big\{w\in\mathcal{D}^{1,p}:\int h|\varphi_1|
 ^{p-2}\varphi_1w=0\big\}
 \end{gathered}
 \end{equation}
In order to prove our main result we use some of the results
introduced by Alziary, Fleckinger and Tak\'{a}\u{c} in
\cite{Takac} where the cases $1<p<2\mbox{ and } 2<p<N$ are treated
separately. The case $2<p<N$  requires the use of the so called
``Improved Poincar\'{e} inequality'' (\cite[lemma 3.7 p.8]{Takac}):
\begin{equation}
\label{ipi}
\begin{gathered}
\int |\nabla u|^p-\lambda_1\int h|u|^p\geqslant
c\Big(|\alpha|^{p-2}\int |\nabla \varphi_1|^{p-2}|\nabla w|^2
+\int|\nabla w|^p\Big),\\
c>0,\quad 2<p<N
\end{gathered}
\end{equation}
where $h$ satisfies the hypothesis:
\begin{itemize}
\item[(H)] The function $h$ is radially symmetric,
$h(x)=h(|x|)$. There exist constants $\delta>0$ and $C>0$ such that
\begin{equation}
\label{H}
0<h(r)\leqslant\frac{C}{(1+r)^{p+\delta}}\mbox{\hspace{3pt}  for
almost all }0\leqslant r<\infty \;\;(r=|x|).
\end{equation}
\end{itemize}
Following  \cite{Takac} we define:
\begin{gather*}
\mathcal{C}_\gamma\stackrel{\text{def}}{=}
\left\{u=\alpha \varphi_1 + w \in\mathcal{D}^{1,p}:\|w\|\leqslant
\gamma |\alpha|   \right\},\\
\mathcal{C'}_\gamma\stackrel{\text{def}}{=}\left\{u\in\mathcal{D}^{1,p}:
\|w\|\geqslant \gamma |\alpha|   \right\},\mbox{ for }0<\gamma<\infty, \\
\mathcal{C'}_\infty\stackrel{\text{def}}{=}
\left\{u\in\mathcal{D}^{1,p}:|\alpha|=0   \right\}
\end{gather*}
The next two lemmas are borrowed from \cite{Takac}
(Lemma 6.2 in page 18, and Lemma 6.3 in page 19).
They play important roles in the proof of  our main result.

\begin{lema} \label{Lambda}
If $h$ satisfies $(H)$, $1<p<N$, and  $0<\gamma\leqslant\infty$ then
\begin{equation}
\label{lambda}
\lambda_1 < \Lambda_\gamma\stackrel{\text{def}}{=}
\inf\Big\{\frac{\int|\nabla u|^p}{\int h|u|^p}:
u\in\mathcal{C'}_\gamma\mbox{\textbackslash} \{0\}\Big\}.
\end{equation} \end{lema}

For the case in which $\|w \|/|\alpha|$ is small  the following lemma
is needed.

\begin{lema} \label{Lambdatilde}
If $h$ satisfies $(H)$ and $2\leqslant p<N$, then
\begin{equation}
\label{lambda tilde}
\lambda_1 < \tilde{\Lambda}\stackrel{\text{def}}{=}
\liminf_{\|\phi \|\to 0,\;\phi\in W} \Big\{\frac{\int|\nabla
(\varphi_1+\phi)|^p}{\int
h|\varphi_1+\phi|^p}:u\in\mathcal{C'}_\gamma\mbox{\textbackslash}
\{0\}\Big\}.
\end{equation}
\end{lema}

The main result of this paper is the following.

\begin{theorem}\label{thm1}
Let  $h$ satisfy $(H)$, $\lambda=\lambda_1$, $g\in
\mathcal{C}(\mathbb{R},\mathbb{R})$ be bounded, $G(s)=\int^s_0
g(t)dt$ and $a\in L^{(p^{*})'}(\mathbb{R}^N)\cap
L^{\infty}(\mathbb{R}^N)$. If
\begin{equation}
\label{alp}
\lim_{|t|\rightarrow\infty}\Big\{\int a(x)G(t\varphi_1)
+t\int f(x)\varphi_1\Big\}=+\infty,
\end{equation}
then,  problem \eqref{eq1} has a weak solution for $2<p<N$.
\end{theorem}

Note that condition \eqref{alp} is a Landesman-Lazer type
condition (see \cite{LL}). Related problems for the $p$-Laplacian
near Resonance had been studied by To Fu Ma et al. \cite{ToFu},
with a different settings for the function $F(x,u):=a(x)g(u)$. The
study for the case $p=2$, without the hypothesis (H), is treated by L\'{o}pez
and Rumbos in \cite{LR}. The existence of weak solutions for
\eqref{eq1} is an extension of previous results for bounded
domains and the ordinary Laplacian by Ahmad, Lazer and Paul
\cite{ALP}.

\noindent\textbf{Remark:} Even though the hypothesis (H) is required for
the proof of our main result, several steps use only that
$h \in L^{N/p}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$.

\section{Variational Setting}

The solutions to (\ref{eq1}) are the critical points of the functional
 \begin{equation}\label{J}
 J_{\lambda}(u)=\frac{1}{p}\int |\nabla u|^p-\frac{\lambda}{p}
 \int h(x)|u|^p-\int aG(u)-\int fu
 \end{equation}
 where  $G(s)=\int_{0}^{s} g(t)dt$, $s\in \mathbb{R}$. It is known
 (see for instance \cite{O}) that the functional $J_{\lambda}$ belongs
 to $\mathcal{C}^1(\mathcal{D}^{1,p},\mathbb{R}^N)\mbox{ for }
u\in\mathcal{D}^{1,p}$ with Fr\'{e}chet derivative given by:
\begin{equation}\label{J'}
\left<J'_{\lambda}(u),v\right>=\int|\nabla u|^{p-2}\nabla u\cdot\nabla v -\lambda\int h|u|^{p-2}uv-\int ag(u)v-\int fv
\end{equation}
for all $u,v\in\mathcal{D}^{1,p}$.

To prove theorem \ref{thm1} we use the Minimax Methods introduced
 by Rabinowitz \cite{R}. We recall here for the convenience of the reader
some previous definitions and theorems.

\noindent{\bf Palais-Smale condition.} Suppose that $E$ is a real Banach
space. A functional $I\in\mathcal{C}^1(E,\mathbb{R})$ satisfies the
Palais-Smale condition at level $c\in\mathbb{R}$, denoted $(PS)_c$, if
any sequence $(u_n)\subset E$ for which
\begin{itemize}
\item[(i)] $I(u_n)\to c$ as $n\to\infty$ and
\item[(ii)] $I'(u_n)\to 0$ as $n\to\infty$,
\end{itemize}
possesses a convergent subsequence. If $I\in\mathcal{C}^1(E,\mathbb{R})$
satisfies the $(PS)_c$ for every $c\in\mathbb{R}$, we say that
$(u_n)$ satisfies the $(PS)$ condition. Any sequence for which (i)
and (ii) hold is called a $(PS)_c$ sequence for $I$.

Now we establish a preliminary result.

\begin{proposition}\label{propPS}
Let $J_{\lambda}:\mathcal{D}^{1,p}\rightarrow\mathbb{R}$ be
defined as \ref{J} where $\lambda\in \mathbb{R}$. Suppose that $g$
is a continuous function with $|g(s)|\leqslant M$ for all $s\in
\mathbb{R}$, $f\in L^{(p^{*})'}(\mathbb{R}^N)$, $2<p<N$,
$h\in L^{N/p}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$,
$h\geqslant 0$  a.e. and
$a\in L^{(p^{*})'}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$.
Then if every $(PS)_c$ sequence for
$J_{\lambda}$ is bounded, $J_\lambda$ satisfies the $(PS)_c$
condition.
\end{proposition}

\begin{proof}
In the first place we note that if $h$ satisfies (H) then
$h\in L^{N/p}(\mathbb{R}^N)$. In fact,
\[
\int h(x)^{N/p}dx=\int_0^{\infty} h(r)^{N/p}r^{N-1}dr
\leqslant \int_0^{\infty}\big(\frac{C}{(1+r)^{p+\delta}}\big)^\frac{N}{p}
r^{N-1}dr <\infty.
\]
 Let $(u_n)$ be a $(PS)_c$ sequence for $J_\lambda$. Thus, by assumption $(u_n)$ is bounded, therefore there exists a subsequence, which we also denote by $(u_{n})$ such that $u_{n}\to u$ weakly in $\mathcal{D}^{1,p}\mbox{ as }n\to\infty$, in particular we have
\begin{equation}
\label{PS1}
\int |\nabla u_{n}|^{p-2}\nabla u_{n}\cdot \nabla\varphi\to\int|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi, \;\;\forall\varphi\in\mathcal{D}^{1,p}.
\end{equation}
Passing to a subsequence if necessary, we see
 that $\int |\nabla(u_{n}-u)|^{p}\to 0\mbox{ as }n\to\infty$. Now
since $(u_{n})$ satisfies the $(PS)_c$ condition,
$\lim_{n\to\infty}\langle J'_{\lambda}(u_n),\varphi\rangle=0$.
 That is,
\begin{equation} \label{PS2}
\int |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla\varphi
-\lambda\int h|u_n|^{p-2}u_n\varphi-\int ag(u_n)\varphi-\int f\varphi=o(1)
\end{equation}
as $n\to\infty$. On the other hand, by weak convergence, we obtain
$$
\lim_{n\to\infty}\int |\nabla u|^{p-2}\nabla u\cdot \nabla(u_n-u)=0.
$$
For $p>2$  (see \cite{All} inequality (7) p.237 and subsequent inequalities)
\begin{equation} \label{alegreto}
\begin{aligned}
\int|\nabla u_n-\nabla u|^p
&\leqslant C\left\{\int (|\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}
\nabla u)\cdot \nabla(u_n-u) \right\}\\
&\quad \times \Big(\int|\nabla u_n|^p+  \int|\nabla u|^p \Big).
\end{aligned}
\end{equation}
Thus it is sufficient to show that
$\lim_{n\to\infty}\int |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla(u_n-u)=0$.
To this aim, taking $\varphi=u_n-u$, in \eqref{PS2} we have
\begin{equation} \label{PS3}
\begin{aligned}
&\int |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla(u_n-u)\\
&=\lambda\int h|u_n|^{p-2}u_n(u_n-u)+\int ag(u_n)(u_n-u)
+\int f(u_n-u)+o(1)
\end{aligned}
\end{equation}
 as $n\to\infty$.
For the first integral in the right hand side, using the H\"{o}lder's
inequality we have
$$
\big|\int h|u_n|^{p-2}u_n(u_n-u)\big|\leqslant
\Big(\int h|u_n|^p\Big)^{1/p'}\Big( \int h|u_n-u|^p\Big)^{1/p}.
$$
Noting that $h\in L^{N/p}(\mathbb{R}^N)=L^{(p^{*}/p)'}(\mathbb{R}^N)$, for
$1\leqslant q<  p^{*}$ the functional $u\mapsto\int h|u|^q$ is
weakly continuous in $\mathcal{D}^{1,p}$ (see \cite[Prop. 2.1 p. 826]{Ben}).
Consequently,
\begin{equation}\label{PS4}
\lim_{n\to\infty} \int h|u_n|^{p-2}u_n(u_n-u)=0.
\end{equation}
For the integral $\int a g(u_n)|u_n-u|$ we consider the ball
$B_r(0)$. Since $a,\, g$ are bounded we have
$$\big|\int_{B_r(0)} ag(u_n)(u_n-u)\big|\leqslant
C\int_{B_r(0)}|u_n-u|\to 0 \quad \mbox{as }n\to\infty,
$$
since $u_n\to u$ strongly in $L^{1}(B_r(0))$ due to the Relich-Kondrachov
theorem. Now, together with the assumption that $u_n\mbox{ and }g$
are bounded, we obtain
$$
\big|\int_{\mathbb{R}^N\setminus B_r(0)}ag(u_n)|u_n-u|\big|\leqslant
 C\Big( \int_{\mathbb{R}^N\setminus B_r(0)}|a|^\frac{Np}{N-p}\Big)
 ^\frac{N+p}{Np}
$$
So, by taking $r$ big enough it follows that
$$
\limsup_{n\to\infty}\big|\int ag(u_n)|u_n-u|\big|\leqslant C\varepsilon
$$
For arbitrary $\varepsilon$. Finally, since $f\in L^{(p^{*})'}(\mathbb{R}^N)$
we can use similar arguments as above to show that
$\lim_{n\to\infty}\int f(u_n-u)=0$.
\end{proof}

\section{Proof of Theorem \ref{thm1}}

In this section we consider the problem
\begin{equation} \label{pl1}
\begin{gathered}
-\Delta_p u=\lambda_1h(x)|u|^{p-2}u+a(x)g(u)+f(x)\\
u\in\mathcal{D}^{1,p}
\end{gathered}
\end{equation}
where $h$ satisfies (H).
To prove the main theorem of this section we require the
Saddle Point Theorem of Rabinowitz \cite{R}, which we introduce
here for the convenience of the reader.

\begin{theorem}[Saddle Point Theorem] \label{sadd}
 Let $E=V\oplus W $, where $E$ is a real Banach space and $V\not= \{0\}$ is
finite dimensional. Suppose $I\in \mathcal{C}^1(E,{\mathbb{R}})$ satisfies the
(PS) condition and
\begin{itemize}
\item[(I1)] there is a constant $\alpha$ and a bounded neighborhood
$D$ of 0 in $V$ such that $I\big|_{\partial D}\leqslant\alpha$, and
\item[(I2)] there is a constant $\beta>\alpha$ such that
$I|_{W}\geqslant \beta$.
\end{itemize}
Then, $I$ possesses a critical value $c\geqslant\beta$. Moreover $c$ can be
characterized as
$$
c=\inf_{h\in\Gamma}\max_{u\in\overline{D}} I(h(u)),
$$
where
$\Gamma=\{h\in\mathcal{C}(\overline{D},E):h=\mbox{id on }\partial D\}$.
\end{theorem}

Now, we can show the existence of weak solutions for $J_{\lambda_1}$.

\begin{proof}[Proof of Theorem \ref{thm1}]
First, we show that the functional $J_{\lambda_1}$ corresponding to
problem \eqref{pl1} satisfies the $(PS)_c$ condition for any
$c\in\mathbb{R}$, and thereafter we verify that $J_{\lambda_1}$
satisfies the other hypotheses of the Theorem \ref{sadd}.

Let $(u_n)$ be a $(PS)_c$ sequence for the functional $J_{\lambda_1}$.
We claim that $(u_n)$ is bounded. For each $n\in\mathbb{N}$ write
$$
u_n\stackrel{\text{def}}{=}v_n+w_n=\alpha_n\varphi_1+w_n\quad \mbox{with }
\alpha_n\in\mathbb{R}\mbox{ and }w_n\in W.
$$
Since $(u_n)$ is a $(PS)_c$ sequence we have $|J_{\lambda_1}(u_n)|<c$, i.e.
\begin{equation}
\label{t1.1}
\Big|\frac{1}{p}\int |\nabla u_n|^p-\frac{\lambda_1}{p}
\int h|u_n|^p-\int aG(u_n)-\int fu_n\Big|\leqslant C_1.
\end{equation}
By inequality \eqref{ipi}, we have
\begin{equation} \label{t1.2}
\frac{c}{p}\int |w_n|^p\leqslant\big|\frac{1}{p}\int |\nabla u_n|^p
-\frac{\lambda_1}{p}\int h|u_n|^p\big|
\end{equation}
with $c>0$. By standard calculations (see for instance \cite[p.16]{LR}),
we have
\begin{equation}
\label{t1.3}
 \big|\int a(G(v_n +w_n)-G(v_n))\big|\leqslant M\int a|w_m|
\leqslant C_2\|w_n\|.
 \end{equation}
Consequently, using \eqref{t1.1}, \eqref{t1.2} and \eqref{t1.3} we have
\begin{equation}
\label{t1.4}
\big|\int aG(v_n)+\int fv_n\big| \leqslant C_1+C_2\|w_n\|+\frac{c}{p}\|w_n\|^p.
\end{equation}
So, given that
$\int aG(v_n)+\int fv_n\to \infty\mbox{ as }\|v_n\|=|\alpha_n|\to\infty$,
we have shown that $(v_n)$ is bounded if $(w_n)$ is bounded. We show now
that $(w_n)$ is bounded. In fact, note that
\begin{equation}
\label{t1.5}
\int |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla w_n
\geqslant \|u_n\|^p-\int |\nabla u_n|^{p-2}u_n\cdot \nabla v_n.
\end{equation}
On the other hand, since
$\langle J'_{\lambda_1}(u_n),v_n \rangle\stackrel{n}{\rightarrow} 0$,
there exists $m_0$ such that if $n\geqslant m_0$ then,
\begin{equation} \label{t1.6}
\big|\int |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla w_n-
\int z_n \cdot w_n\big|\leqslant C\|w_n\|,
\end{equation}
where $z_n =  \lambda_1 h|u_n|^{p-2}u_n + ag(u_n)w_n+  f$.
Adding and subtracting $\lambda_1\int h|u_n|^{p-2}u_n\cdot
v_n\mbox{ and }\int ag(u_n)v_n+\int f(x)v_n$, and substituting
\eqref{t1.5} in \eqref{t1.6}
\begin{equation}\label{t1.7}
\begin{aligned}
\|u_n\|^{p}-\lambda_1\int h|u_n|^p
&\leqslant C\|w_n\|+\langle J'_{\lambda_1}(u_n),v_m\rangle+
\int ag(u_n)v_n+\int f(x)v_n\\
&\leqslant C\|w_n\|+\langle J'_{\lambda_1}(u_n),v_m\rangle+C'|\alpha_n|
\end{aligned}
\end{equation}
Again, since $J'_{\lambda_1}\to 0\mbox{ as }n\to\infty$, there exist
$m_1$ such that if
$n\geqslant m_1$ then $\langle J'_{\lambda_1}(u_n),v_m\rangle
\leqslant C\|v_n\|=C|\alpha_n|$,
taking $n\geqslant\max\{m_0,m_1\}$
\begin{equation}\label{t1.8}
\|u_n\|^{p}-\lambda_1\int h|u_n|^p\leqslant C\|w_n\|+C'|\alpha_n|.
\end{equation}
Now fix $\gamma>0$, and suppose that $(u_n)\in \mathcal{C'}_\gamma$ for all $n$. Then we have, $|\alpha_n|\leqslant (1/\gamma)\|w_n\|$ and $\int h|u_n|^p\leqslant(1/\Lambda_{\gamma})\int|\nabla u_n|^p$. Thus, by Lemma \ref{Lambda}
\begin{equation}
\label{t1.9}
\big(1 -\frac{\lambda_1}{\Lambda_{\gamma}}\big)\|u_n\|^p\leqslant C\|w_n\|
\end{equation}
Since the projection $u\mapsto w$ is bounded in $\mathcal{D}^{1,p}$ we obtain
\begin{equation}
\label{t1.9'}
\|w_n\|^p\leqslant C_\gamma\|w_n\|,
\end{equation}
given that $\lambda_1/ \Lambda_\gamma<1$ by Lemma \ref{Lambda}.

Hence by Lemma \ref{Lambda}, $\Lambda_\gamma>\lambda_1$; therefore,
 $(w_n)$ is bounded if $(u_n)\in\mathcal{C'}_\gamma$.
Now, set $\gamma_n =\|w_n\|/|\alpha_n|$ and define
$$
\gamma \stackrel{\text{def}}{=}\liminf_n\gamma_n.
$$
We have two cases:
(i) $\gamma\in(0,\infty]$ and (ii) $\gamma=0$.
By the above  argument, if $\gamma\in(0,\infty]$ then $(w_n)$
is bounded and the proof is concluded. If $\gamma=0$, take $\varepsilon>0$
arbitrarily small, such that $\|w_n\|\leqslant\varepsilon|\alpha_n|$.
Using inequality \eqref{t1.8}, Lemma \ref{Lambdatilde} with
$\phi=\phi_n \stackrel{\text{def}}{=}(\|w_n\|/|\alpha_n|)\cdot w_n/\|w_n\|$,
and the fact that the projection $u\mapsto \alpha$ is bounded in
$\mathcal{D}^{1,p}$ we obtain
\begin{gather*}
 |u_n|^p\big( 1-\frac{\lambda_1}{\tilde{\Lambda}}  \big)
 \leqslant C\varepsilon|\alpha_n|+C'\|v_n\|,\\
|\alpha_n|^p\leqslant c_\gamma|\alpha_n|.
\end{gather*}
Therefore, $|\alpha_n|$  is bounded, and since $\|w_n\|\leqslant\varepsilon|\alpha_n|$ we have that $(u_n)$ is bounded as wanted.

To verify the geometric hypotheses of the Saddle Point Theorem we note that since $\lambda_1$ is isolated (see \cite{All}) we have
\begin{equation}
\label{g1}
\lambda_2\stackrel{\text{def}}{=}\inf\big\{\|w\|^p:w\in W,\int h|w|^p=1 \big\},
\end{equation}
which satisfies $\lambda_1<\lambda_2$. As a consequence of \eqref{g1} we have
\begin{equation} \label{g2}
\int|\nabla w|^p\geqslant\lambda_2\int h|w|^p,\quad \forall w\in W.
\end{equation}
Now, if $w\in W$,
\begin{equation} \label{g3}
\int|\nabla w|^p-\lambda_1\int h|w|^p\geqslant
\big(1-\frac{\lambda_1}{\lambda_2}\big).
\end{equation}
Moreover, since $|g(s)|\leqslant M$ for all $s\in\mathbb{R}$, we have that
 for all $w\in\mathcal{D}^{1,p}$,
$$
\big|\int aG(w) \big|\leqslant M\int |a||w|\leqslant C\|w\|.
$$
Therefore, $J_{\lambda_1}$ is bounded from below on $W$; i.e.
(I2) in Theorem \ref{sadd} holds.

Finally, if $v\in V$ we have
$$
J_{\lambda_1}(v)=-\int aG(v)-\int fv.
$$
Since $\int aG(v)+\int fv\to\infty$  as $\|v\|\to\infty$ by \eqref{alp}
and, therefore, (I1) in the Saddle Point Theorem also holds.
Hence, $J_{\lambda_1}$ has a critical point and the proof is concluded.
\end{proof}

\noindent\textbf{Remark.}
 Suppose $\lim_{s\to\infty} g(s)=g_{\infty}$ and
$\lim_{s\to-\infty} g(s)=g_{-\infty}$ exist. Then, if
$g_{\infty}>0$ and $g_{-\infty}<0$,
$G(s)=\int^{s}_{0}g(t)dt\to\infty$ as
$|s|\to\infty$. Consequently, by L' H\^{o}spital's rule,
the Lebesgue dominated convergence theorem and the fact that
$\varphi_1>0$ a.e. in $\mathbb{R}^N$ we have that
$$
\lim_{|t|\to\infty}\frac{1}{t}\int a(x)G(t\varphi_1)
=\lim_{|t|\to\infty}\int ag(t\varphi_1)\varphi_1
= \begin{cases}
g_{\infty}\int a\varphi_1 &\mbox{as }t\to\infty,\\
g_{-\infty}\int a\varphi_1 &\mbox{as }t\to-\infty .
\end{cases}
$$
Thus, the condition \eqref{alp} in the resonance Theorem \ref{thm1} holds if
\[
g_{\infty}\int a\varphi_1+\int f\varphi_1>0\quad
\mbox{and}\quad g_{-\infty}\int a\varphi_1+\int f\varphi_1<0 ,
\]
or
\begin{equation} \label{LL}
g_{-\infty}\int a\varphi_1<-\int f\varphi_1<g_{\infty}\int a\varphi_1.
\end{equation}
This is the original Landesman-Lazer condition in \cite{LL} for the case
of resonance around the first eigenvalue.

It can be shown that if
$$
g_{-\infty}<g(s)<g_{+\infty}\quad\mbox{for all }s\in\mathbb{R},
$$
then \eqref{LL} is necessary and sufficient for the solvability
of \eqref{pl1}. If $g_{-\infty}=g_{+\infty}$, then the
Landesman-Lazer condition \eqref{LL} cannot hold, and if
$g_{-\infty}$ and $g_{+\infty}$ are both zero, then
condition \eqref{alp} might not hold in general.


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\end{document}