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%\input amssym.def     % used for R in Real numbers
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\markboth{\hfil Perturbed eigenvalue problems \hfil EJDE--1997/11}%
{EJDE--1997/11\hfil Jo\~{a}o Marcos B. do \'{O}\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1997}(1997), No.\ 11, pp. 1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
Solutions to perturbed eigenvalue problems of the $p$-Laplacian in
${\Bbb R}^N$ 
\thanks{ {\em 1991 Mathematics Subject Classification:}
35A15, 35J60.\hfil\break\indent
{\em Key words and phrases:} Elliptic Equations on unbounded Domains,
$p$-Laplacian, \hfil\break\indent
Mountain Pass Theorem, Palais-Smale Condition, First eigenvalue, 
\hfil\break\indent
\copyright 1997 Southwest Texas State University  and University of
North Texas.\hfil\break\indent
Work partially supported by CNPq/Brazil.\hfil\break\indent
Submitted January 24, 1997. Published July 15, 1997.} }
\date{}
\author{Jo\~{a}o Marcos B. do \'{O}}
\maketitle
\begin{abstract} 
Using a variational approach, we investigate the existence of solutions 
for  non-autonomous perturbations of the 
$p-$Laplacian eigenvalue problem 
$$
 -\Delta _pu=f(x,u)\quad \mbox{in}\quad {\Bbb R}^N\,. 
$$
Under the assumptions that the primitive $F(x,u)$ of $f(x,u)$
interacts only with the first eigenvalue, we look for solutions in
the space $D^{1,p}({\Bbb R}^N)$. 
Furthermore, we assume a condition that measures how different the 
behavior  of the function $F(x,u)$ is from that of the $p-$power of $u$.
\end{abstract} 

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newcommand{\diver}{\mathop{\rm div}}
\newcommand{\supp}{\mathop{\rm supp}}

\section{Introduction}
In this paper we study the existence of solutions for 
non-autonomous perturbations of the $p$-Laplacian eigenvalue problem
\begin{equation}
-\Delta _pu\equiv -\diver\left( |\nabla u|
^{p-2}\nabla u\right) =f(x,u)\quad\mbox{in}\quad {\Bbb R}^N\,, \label{1}
\end{equation}
where $u\in D^{1,p}({\Bbb R}^N)$. 
We assume that the primitive $F(x,u)$ of
the nonlinearity $f(x,u)$ interacts only with the first eigenvalue of some 
$p$-Laplacian eigenvalue problems with weights naturally associated with 
$F(x,u)$. We also assume that $f:{\Bbb R}^N\times {\Bbb R}\rightarrow 
{\Bbb R}$ is a continuous function satisfying the growth condition,
$$
|f\left( x,u\right)|\leq a(x)|u|^r+b(x)|u|^s,\quad 
\forall(x,u)\in {\Bbb R}^N\times {\Bbb R} \leqno{(f)}
$$
where $a,b$ are continuous functions, 
$a\in L^\infty ({\Bbb R}^N)\cap L^{r_0}({\Bbb R}^N)$ and 
$b\in L^\infty ({\Bbb R}^N)\cap L^{s_0}({\Bbb R}^N)$ with
$0\leq r\leq p-1\leq s<p^{*}-1$, $r_0=Np/[Np-(r+1)(N-p)]$, and 
$s_0=Np/[Np-(s+1)(N-p)]$. In this context $1<p<N$, and $p^{*}$ denotes 
the critical Sobolev exponent, $p^{*}=Np/(N-p)$. 

We are interested in finding weak solutions of (\ref{1})
 in the framework of the reflexive Banach space $D^{1,p}({\Bbb R}^N)$, 
defined as the completion of $C_0^\infty ({\Bbb R}^N)$ with
respect to the norm $\| u\| _{D^{1,p}}=\| \nabla u\| _{L^p}$. 
By the Sobolev inequality, 
$\| u\|_{L^{p^{*}}}\leq C_0\| \nabla u\| _{L^p}$ for all $u\in $ $%
D^{1,p}({\Bbb R}^N)$, we see that $D^{1,p}({\Bbb R}^N)$ can be embedded
continuously in $L^{p^{*}}({\Bbb R}^N)$ and that
\[
D^{1,p}({\Bbb R}^N)=\{u\in L^{p^{*}}({\Bbb R}^N):\nabla u\in (L^p({\Bbb R}%
^N))^N\}\,. 
\]
For more details about this space, see e.g. \cite{Ben-Troestler-Willem}. As
we shall see, under assumption $(f)$, the functional 
\[
I\left( u\right) =\frac 1p\int_{{\Bbb R}^N}|\nabla u|^p\,dx-\int_{%
{\Bbb R}^N}F(x,u)\,dx,\quad F(x,u)=\int_0^uf(x,s)\,ds, 
\]
is a $C^1$ weakly lower semi-continuous functional defined on $D^{1,p}({\Bbb R%
}^N)$, and
\[
I' \left( u\right) \varphi =\int_{{\Bbb R}^N}|\nabla u|
^{p-2}\nabla u\nabla \varphi dx-\int_{{\Bbb R}^N}f(x,u)\varphi dx,\quad
\forall \varphi \in D^{1,p}({\Bbb R}^N). 
\]
Notice that critical points of the functional $I$ are precisely the weak
solutions of (\ref{1}). To get critical points of $I$ we apply two
``minimax'' methods: a mountain pass type argument and
a minimization technique. To do this, we explore the interaction of the
potential $F(x,u)$ with the first eigenvalue, in combination with the
following hypotheses which are related with a compactness condition of
Palais-Smale type.
\begin{description}
\item{$(F_1)$} There exists a measurable function $a\in L^\infty (%
{\Bbb R}^N)$ and a constant $\mu$, $0<\mu <p^*$ such that 
\[
f(x,u)u-pF(x,u)\geq a(x)|u|^\mu >0,\quad \forall (x,u)\in {\Bbb R}%
^N\times ({\Bbb R-\{}0\}), 
\]

\item{$(F_2)$} For some $p<q<p^*$, $\limsup\limits_{|u|\rightarrow
+\infty }\frac{F(x,u)}{a(x)|u|^q}\leq M<+\infty $ uniformly 
$x\in {\Bbb R}^N$
\end{description}

Let $\lambda _1(m)$ denote the first eigenvalue of the weighted nonlinear
eigenvalue problem in ${\Bbb R}^N$, 
\[
-\Delta _pu=\lambda m(x)|u|^{p-2}u, 
\]
where $m\in L^{N/p}({\Bbb R}^N)$ is a weight function which is positive on a
subset of positive measure. It is convenient to recall the
variational characterization 
\begin{equation}
\lambda _1(m)=\inf \{\int_{{\Bbb R}^N}|\nabla u|^pdx/\int_{{\Bbb R}%
^N}m|u|^pdx:u\in D^{1,p}({\Bbb R}^N)-\{0\}\}\,, \label{A1}
\end{equation}
and that $\lambda _1(m)$ is a positive real number, since by the Holder
and Sobolev inequalities
\[
\int_{{\Bbb R}^N}m|u|^pdx\leq \| m\| _{L^{N/p}}\| u\| _{L^{p^{*}}}^p
\leq C\| m\| _{L^{N/p}}\| u\| _{D^{1,p}}^p. 
\]
Thus, 
\[
\frac{\int_{{\Bbb R}^N}|\nabla u|^pdx}{\int_{{\Bbb R}^N}m|u|
^pdx}\geq \frac 1{C\| m\| _{L^{N/p}}^{}}>0, 
\]
for all $u\in D^{1,p}({\Bbb R}^N)-\{0\}$.
For more details about this eigenvalue problem, see e. g. 
\cite{Allegretto-Huang}.

Now, we are ready to present the main results of this article.

\begin{theorem}
Assume that $(f)$, $(F_1)$, and $(F_2)$ are satisfied. Furthermore, 
suppose 
\begin{description}
\item{($F_3$)}  There exist a function $\alpha \in L^\infty ({\Bbb R%
}^N)\cap L^{\frac Np}({\Bbb R}^N)$ and a positive Conant $\delta$, 
such that 
\[
F(x,u)\leq \frac 1p\alpha (x)|u|^p\quad \forall x\in {\Bbb R}^N,\
\forall |u|\leq \delta , 
\]
where either $\alpha \leq 0$ or $\lambda _1(\alpha )>1$,
\item{($F_4$)} There exist a function $\omega \in L^\infty ({\Bbb R}%
^N)\cap L^{\frac Np}({\Bbb R}^N)$ and a positive constant $R$, 
such that 
\[
F(x,u)\geq \frac 1p\omega (x)|u|^p\quad \forall x\in {\Bbb R}^N,\
\forall |u|\geq R, 
\]
where $\omega>0$ on a subset of positive measure and 
$\lambda_1(\omega )<1$.
\end{description}
Then, problem (\ref{1})\thinspace has a nontrivial solution, provided
that $0<N(q-p)/p<\mu <q<p^{*}$.
\end{theorem}

Next, we consider the case when the potential $F(x,u)$ approaches
 $\omega (x)|u|^p$ as $|u|\rightarrow \infty $,
and therefore, it interacts with the first eigenvalue of the problem $%
-\Delta _pu=\lambda \omega (x)|u|^{p-2}u$. In fact, we assume that $%
F(x,u)$ interacts with the first eigenvalue $\lambda _1(\omega )$,
 but its behavior is different from that of a $p-$power of $u$. As we shall see,
these facts lead to a standard situation for a global minimum.

\begin{theorem}
Assume that $(f)$ and $(F_1)$ hold. Furthermore, suppose
\begin{description}
\item{$(F_5)$} There is a function $\omega \in L^\infty ({\Bbb R}^N)\cap L^{\frac Np}({\Bbb %
R}^N)$ such that 
\[
\limsup_{|u|\rightarrow +\infty }\frac{pF(x,u)}{|u|
^p}\leq \omega (x)\quad uniformly\ x\in {\Bbb R}^N, 
\]
where either $\omega $ is $\leq 0$ or $\lambda _1(\omega )\geq 1$.
\end{description} 
Then, problem (\ref{1}) has a solution which is a global minimum of the
associated functional, provided that $0<\mu <p.$
\end{theorem}

\paragraph{Remark 1.}
If in Theorem 1, assumption $(F_1)$ is replaced by its ``dual'',
$$f(x,u)u-pF(x,u)\leq -a(x)|u|^\mu <0,\quad
\forall (x,u)\in {\Bbb R}^N\times ({\Bbb R-\{}0\})\,,
\leqno{(\widetilde{F}_1)}$$
it is not difficult to see that similar result holds. 
Hypotheses of this kind are a measure of how different the behavior of $F$ 
is from that of a $p-$power of $u$.
 It is worth to observe that in many cases condition $(F_1)$ is implied
 by the following Ambrosetti-Rabinowitz condition,
\begin{description}
\item{$(AR)$}\quad 
$0<\theta F(x,u)\leq uf(x,u)$, for all $(x,u)\in {\Bbb R}^N\times 
({\Bbb R-\{}0\})$ and some $\theta >p$.
\end{description}
Indeed, by $(AR)$, $f(x,u)u-pF(x,u)\geq (\theta -p)F(x,u)$ and, moreover, $F(x,u)\geq \min
\{F(x,1),F(x,-1)\}|u|^\theta >0,\quad \forall x\in {\Bbb R}^N$ and $%
|u|\geq 1$. Thus, $(F_1)$ holds if we assume also that there exist $%
\mu \leq \theta $ and a measurable function $a\in L^\infty ({\Bbb R}^N)$
such that $F(x,u)\geq a(x)|u|^\mu >0$, for all $x\in {\Bbb R}^N$ and 
$|u|\leq 1$. 
On the other hand, Example~1 shows a function which satisfies 
condition \thinspace $(F_1)$, but does not satisfy $(AR)$. 
Requirements similar to $(F_1)-(F_2)$ were considered by
Costa and Miyagaki in \cite{Costa-Miyagaki} (see also \cite
{Costa-Magalhaes1, Costa-Magalhaes2}) where $a$ is constant.
They obtained a nontrivial solution for autonomous
perturbations of the $p$-Laplacian on a unbounded cylindrical domain,
$\Omega =\Omega _0\times {\Bbb R}^{N-K}$, with $\Omega _0$ bounded domain
of ${\Bbb R}^K$.  Their solution, in the space $W_0^{1,p}(\Omega )$, is
obtained using the mountain pass argument combined with the 
concentration-compactness principle of Lions. 

In order to get the geometry of the mountain-pass, the number 
\[
\lambda _1=\inf \{\int_\Omega |\nabla u|^pdx/\int_\Omega |
u|^pdx:u\in W_0^{1,p}(\Omega )-\{0\}\}, 
\]
has been explored, and it is known (\cite{Burton,Esteban}) to be equal to
the first eigenvalue of $-\Delta _p$ acting on $W_0^{1,p}(\Omega _0)$. 
In fact, the following condition for crossing the first
eigenvalue has been used,
$$\limsup\nolimits_{u\rightarrow 0}pF(u)/|u|
^p\leq \xi <\lambda _1<\eta \leq \liminf\nolimits_{|u|
\rightarrow \infty }pF(u)/|u|^p\,.$$
For the non-autonomous perturbations of the $p$-Laplacian studied here, 
the crossing of the first eigenvalue is expressed by requirements such as $(F_3)$
and $(F_4)$. These conditions give
 the geometric shape required by the Mountain Pass Theorem. Condition 
$(F_3)$ together with the growth condition $(f)$ lead to the fact that the
origin is a local minimum of the associated functional, while assumption $%
(F_4)$ implies that the functional is not bounded below. 
The interaction of the potential $F(x,u)$ with the
first eigenvalue in  elliptic eigenvalue problems with weights
in bounded domains have been considered by de Figueiredo and 
Massab\'{o} in \cite{Figueiredo-Massabo}.


\paragraph{Remark 2.}
The same procedures as in Theorem 1, along with obvious
modifications, show the existence of a nontrivial solution to the 
non-autonomous perturbation of the $p$-Laplacian studied by Yu in 
\cite{Yu}: 
\[
l(u)\equiv -\diver(\left( a(x)|\nabla u|^{p-2}\nabla u+b(x)|u|
^{p-2}u\right) =f(x,u)\mbox{ in }\Omega \mbox{ and } u|
_{\partial \Omega }=0, 
\]
where $\Omega $ is a $C^{1,\delta }$ $(0<\delta <1)$ exterior domain in $%
{\Bbb R}^N$, $0<a_0\leq a(x)\in L^\infty (\Omega )\cap C^\delta (\overline{%
\Omega })$, and $0\leq b(x)\in L^\infty (\Omega )\cap C(\Omega )$.
Following this approach we complement Theorem~1 in Yu, which corresponds
to the to ``super-linear case''. 
In \cite{Yu}, a solution was obtained by working in the framework of a
 reflexive Banach space $E$, defined as the
completion of $C_0^\infty (\Omega )$ with respect to the norm $\|
u\| _E^p=\int_\Omega [|\nabla u|^p+w(x)|u|^p]dx$,
where$\quad w(x)=\max \{b(x),(1+|x|)^{-p}\}$. 
From the Hardy-type inequality, $\int_\Omega (1+|x|)^{-p}|u|^pdx\leq 
\int_\Omega [|\nabla u|^p\,dx$ for all $u\in C_0^\infty (\Omega )$ 
it follows that the norm induced by the operator $l$, 
$\| u\| _l^p=\int_\Omega [|a(x)\nabla u|^p+b(x)|u|^p]dx$, 
is equivalent to the norm in $E$.
Therefore, in order to establish conditions of type $(F_3)$ and $(F_4)$ 
 we must explore ``crossing  conditions'' involving the number 
\[
\lambda _1(m)=\inf \{\| u\| _l^p/\int_\Omega m(x)|u|
^pdx:u\in E-\{0\}\}, 
\]
which is positive in view of the Holder and Sobolev inequalities.
\medskip

\paragraph{Remark 3.}
Problems involving unbounded domains have been
studied recently by several authors, among others
 Berestycki and Lions \cite{Berestycki-Lions}, Strauss \cite{Strauss},
and Dr\'{a}bek \cite{Drabek} (see also references therein).

\section{Preliminary Results}

Throughout this work $C$ will denote a generic positive constant.
Results similar to the one in the next lemma have appeared in 
\cite{Yu}, and we sketch its proof here for the benefit of the reader.

\begin{lemma}
\label{Preliminary}Suppose that $f$ is continuous and satisfies $(f)$. Then
the functional $I:D^{1,p}({\Bbb R}^N)\rightarrow {\Bbb R}$ given by 
\[
I(u)=\frac 1p\int_{{\Bbb R}^N}|\nabla u|^pdx-\int_{{\Bbb R}%
^N}F(x,u)dx, 
\]
is well defined and of class $C^1$ with 
\begin{equation}
I' \left( u\right) \varphi =\int_{{\Bbb R}^N}|\nabla u|
^{p-2}\nabla u\nabla \varphi dx-\int_{{\Bbb R}^N}f(x,u)\varphi dx,\quad
\forall \varphi \in D^{1,p}({\Bbb R}^N).  \label{D1}
\end{equation}
Moreover, the functional 
\[
K\left( u\right) =\int_{{\Bbb R}^N}F\left( x,u\left( x\right) \right) dx 
\]
is weakly continuous and $K'$ is compact.
\end{lemma}

\paragraph{Proof.} As a consequence of assumption $(f)$, with the aid of
the Holder and Sobolev inequalities, we see that $I$ and 
$I' (u)$ are well defined on $D^{1,p}({\Bbb R}^N)$.

The idea in the rest of the proof is to use results similar to the ones
known on the case of bounded domains. We also use that the
restriction operator is continuous, that 
$\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))}\rightarrow 0$, and that
$\| b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}\rightarrow 0$ as 
$n\rightarrow \infty$, for any $(r_n)$ increasing and unbounded sequence 
of positive real numbers. 

Let $B(0,r_n)$ denote the ball of radius $r_n$ centered at
the origin of ${\Bbb R}^N$, and let 
$K_n:D^{1,p}(B(0,r_n))\rightarrow {\Bbb R}$
denote the functional given by 
\[
K_n(u)=\int_{B(0,r_n)}F(x,u)dx. 
\]
In view of $(f)$, it is well known that $K_n\in C^1\left( D^{1,p}\left(
B(0,r_n)\right) \right) $ and that
\[
K_n'(u)(v)=\int_{B(0,r_n)}f(x,u)vdx,\quad \forall v\in D^{1,p}\left(
B(0,r_n)\right) . 
\]
Moreover, $K_n$ is weakly continuous and $K_n^{' }(u)$ is compact
(cf. \cite{Figueiredo,Rabinowitz}).

Next we prove that $K$ is weakly continuous. Let $u_k\rightharpoonup u$
weakly in $D^{1,p}({\Bbb R}^N)$ as $k\rightarrow \infty $. 
Using hypothesis $(f)$ and
the Holder and Sobolev inequalities, we obtain 
\begin{eqnarray*}
|K(u_k)-K(u)|&\leq& |K_n(u_k)-K_n(u)|\\
&&+ C\{\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))}(\|
  u_k\| _{D^{1,p}}^{r+1}+\| u\| _{D^{1,p}}^{r+1})\\
&&+ \| b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}(\|
 u_k\| _{D^{1,p}}^{s+1}+\| u\| _{D^{1,p}}^{s+1})\}\,.
\end{eqnarray*}
Given $\epsilon >0$, we choose $r_n$ sufficiently large such that 
\begin{equation}
\max \{\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))},\|
b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}\}<\epsilon /2C\,.  \label{D2}
\end{equation}
On the other hand, since the restriction operator $u\longmapsto u|
_{B(0,r_n)}$ is continuous from $D^{1,p}({\Bbb R}^N)$ into $D^{1,p}(B(0,r_n)$
and $K_n$ is weakly continuous, up to a subsequence, we have 
\begin{equation}
|K_n(u_k)-K_n(u)|<\epsilon /2\,.  \label{D3}
\end{equation}
Combining (\ref{D2}) and (\ref{D3}) we conclude that $K$ is weakly
continuous.
To prove that $I$ $\in C^1\left( D^{1,p}({\Bbb R}^N)\right) $, we must show
that $K\in C^1\left( D^{1,p}({\Bbb R}^N)\right) $ and that
\[
K' (u)v=\int_{{\Bbb R}^N}f(u,x)vdx, 
\]
since the first term in $I$ is $C^1$ and its Fr\'{e}chet
derivative is the first term in (\ref{D1}). For fixed $u\in D^{1,p}({\Bbb R%
}^N)$ and given $\epsilon >0$, we must show that there exists $\delta
=\delta (u,\epsilon )$ such that 
\[
|\int_{{\Bbb R}^N}F\left( x,u+v\right) dx-\int_{{\Bbb R}%
^N}F(x,u)dx-\int_{{\Bbb R}^N}f(u,x)vdx|\leq \epsilon \|
v\| _{D^{1,p}}, 
\]
for all $v\in D^{1,p}({\Bbb R}^N)$ with $\| v\| _{D^{1,p}}\leq
\delta $. Notice that if $\| v\| _{D^{1,p}}\leq 1$, 
\begin{eqnarray*}
\lefteqn{|\int_{{\Bbb R}^N}F\left( x,u+v\right)\,dx-\int_{{\Bbb R}^N}
F(x,u)dx-\int_{{\Bbb R}^N}f(u,x)v\,dx|} && \\
&\leq & |K_n(u+v)-K_n(v)-K_n' (u)v|\\
&&+ C \| v\| _{D^{1,p}}\{\| a\| _{L^{r_0}({\Bbb R}
^N-B(0,r_n))}(\| u_k\| _{D^{1,p}}^r+\| u\|
_{D^{1,p}}^r)\\
&&+ \| b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}(\|
u_k\| _{D^{1,p}}^s+\| u\| _{D^{1,p}}^s)\}\,.
\end{eqnarray*}
Since $a\in L^{r_0}({\Bbb R}^N)$ and $b\in L^{s_0}({\Bbb R}^N)$, as before,
we can choose $r_n$ sufficiently large such that $\max \{\|
a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))},\| b\| _{L^{s_0}(%
{\Bbb R}^N-B(0,r_n))}\}<\epsilon /2C$.
Using that $K_n\in C^1\left(D^{1,p}\left( B(0,r_n)\right) \right) $, 
there exists $\delta =\delta
(u,\epsilon )$ such that, for all $v\in D^{1,p}({\Bbb R}^N)$ with $\|
v\| _{D^{1,p}}\leq \delta $, 
\[
|K_n(u+v)-K_n(v)-K_n' (u)v|\leq \epsilon \|
v\| _{D^{1,p}}, 
\]
and the proof that $I$ is Fr\'{e}chet differentiable is complete.
To prove that $K' $ is continuous we use the estimate
\begin{eqnarray*}
|K' \left( u_k\right) -K' \left( u\right) | 
&\leq& |K_n' \left( u_k\right) -K_n' \left( u\right) |\\
&&+ C\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))}(\|
 u_k\| _{D^{1,p}}^r+\| u\| _{D^{1,p}}^r)\\
&&+ C\| b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}(\|
u_k\| _{D^{1,p}}^s+\| u\| _{D^{1,p}}^s)
\end{eqnarray*}
together with the facts that $K_n' $ is continuous, and that
$\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))}$ and $\| b\| _{L^{s_0}(%
{\Bbb R}^N-B(0,r_n))}$ converge to zero as $n\rightarrow \infty $.

Finally, to prove compactness, we use the diagonal method. Let $%
u_k\rightharpoonup u$ weakly in $D^{1,p}({\Bbb R}^N)$ as $k\rightarrow
\infty $. Since $a\in L^{r_0}({\Bbb R}^N)$ and $b\in L^{s_0}({\Bbb R}^N)$ we
can choose  an increasing and unbounded sequence of positive
real numbers $(r_n)$ such that 
\[
\max \{\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))},\|
b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}\}\leq 1/2n. 
\]
On the other hand, for each natural number $n$, we have a subsequence $%
(u_{kn})$, of $(u_k)$, such that 
\[
\| K_n' (u_{kn})-K_n' (u)\| \leq 1/2n, 
\]
since $K_n'$ is compact. Thus, combining these two estimates, 
\[
\| K' (u_{kn})-K' (u)\| \leq C/n. 
\]
Therefore, the diagonal subsequence $(u_{kk})$ leads to 
$K' (u_{kk})\rightarrow K' (u)$. $\hfill\Box$

In the next lemmas we prove that the functional $I$ satisfies a compactness
condition of the Palais-Smale type which was introduced by Cerami in \cite
{Cerami}.

\paragraph{Definition}
Let  $J:E\rightarrow {\Bbb R}$ be  a $C^1$ functional. We say that 
$J$ satisfies condition $(C)$ if every sequence $(u_n)$ in $E$ such that 
\begin{equation}
\begin{array}{lll}
(i) &  & J(u_n)\rightarrow c \\ 
(ii) &  & (1+\| u_n\| _{D^{1,p}})\|J'(u_n)\|_{(D^{1,p})^{*}}\rightarrow 0,
\end{array}
\label{C1}
\end{equation}
possesses a convergent subsequence.
\medskip

Using this compactness condition, Bartolo, Benci and Fortunato in \cite
{Bartolo-Benci-Fortunato} obtained rather general minimax results.

\begin{lemma}[Compactness Condition I]
\label{Compactness1} Assume that $(f)$, $(F_1),\ (F_2)$, and $(F_3)$ 
are satisfied with $0<N(q-p)/p<\mu <q<p^{*}$. Then, the functional $I$ 
satisfies the compactness condition $(C).$
\end{lemma}

\paragraph{Proof.} Let $(u_n)\,\subset D^{1,p}({\Bbb R}^N)$ be a sequence
satisfying (\ref{C1}). From $(F_1)$ together with (\ref{C1}), we have 
\begin{eqnarray*}
C &\geq &pI(u_n)-I' (u_n)u_n \\
&=&\int_{{\Bbb R}^N}[f(x,u_n)u_n-pF(x,u_n)]dx \\
&\geq &\int_{{\Bbb R}^N}a(x)|u_n|^\mu dx,
\end{eqnarray*}
for some positive constant $C$ and for sufficiently large $n$. Thus, $(u_n)$
is a bounded sequence in the Lebesgue space 
\[
L_a^\mu ({\Bbb R}^N)=\{u:\int_{{\Bbb R}^N}a(x)|u_n|^\mu dx<\infty
\}. 
\]
From \cite{Rudin}, we have the interpolation inequality
\[
\| u\| _{L_a^q}\leq \| u\| _{L_a^\mu
}^{1-t}\| u\| _{L_a^{p^{*}}}^t,\quad \forall u\in L_a^\mu (%
{\Bbb R}^N)\cap L_a^{p^{*}}({\Bbb R}^N), 
\]
where $0<\mu <q<p^{*}$ and $t\in (0,1)$ are such that $1/q=(1-t)/\mu
+t/p^{*}$.  The previous two inequalities and 
 $a\in L^\infty ({\Bbb R}^N)$ imply 
\begin{equation}
\| u_n\| _{L_a^q}^{}\leq C\| u_n\|
_{L_{}^{p^{*}}}^t.  \label{C2}
\end{equation}
Using $(F_2)$ and $(F_3)$ we obtain 
\[
F(x,u)\leq \frac{\alpha (x)}p|u|^p+Ca(x)|u|^q,\ \forall
(x,u)\in {\Bbb R}^N\times {\Bbb R}. 
\]
Thus, from $(i)$ in (\ref{C1}), we achieve 
\begin{eqnarray*}
\int_{{\Bbb R}^N}|\nabla u_n|^pdx &=& pI(u_n)+p\int_{{\Bbb R}^N}F(x,u_n)\,
 dx \\ 
&\leq& C+\int_{{\Bbb R}^N}\alpha (x)|u_n|^pdx+C\int_{{\Bbb R}%
^N}a(x)|u_n|^q\,dx\,,
\end{eqnarray*}
which, in view of the variational characterization of the first eigenvalue $%
\lambda _1(\alpha )$ and (\ref{C2}), guarantees that 
\begin{eqnarray*}
\| u_n\| _{D^{1,p}}^p &\leq& C+\lambda _1^{-1}(\alpha
)\| u_n\| _{D^{1,p}}^p+C\| u_n\| _{L^{p^{*}}(%
{\Bbb R}^N)}^{tq} \\ 
& \leq& C+\lambda _1^{-1}(\alpha )\| u_n\|
_{D^{1,p}}^p+C\| u_n\| _{D^{1,p}}^{tq}\,,
\end{eqnarray*}
where, in the last estimate we used the Sobolev inequality. Thus, 
\[
(1-\lambda _1^{-1}(\alpha ))\| u_n\| _{D^{1,p}}^p\leq
C(1+\| u_n\| _{D^{1,p}}^{tq}). 
\]
Therefore, due to $\lambda _1(\alpha )>1$, the sequence $(u_n)$
is bounded, since $\mu >\frac Np(q-p)$ is equivalent to $tq<p$.

Finally, by considering the functional $J:D^{1,p}({\Bbb R}^N)\rightarrow 
{\Bbb R};$ $J(u)=\frac 1p\| u_n\| _{D^{1,p}}^p$, whose
derivative, $J' :D^{1,p}({\Bbb R}^N)\rightarrow J' (D^{1,p}(%
{\Bbb R}^N))$, $J' (u)\varphi =\int |\nabla u|^{p-2}\nabla
u\nabla \varphi $, is an isomorphism, we have 
\[
(J' )^{-1}I' (u)=u-(J' )^{-1}K' (u). 
\]
This fact together with $\| I' (u_n)\|
_{(D^{1,p})^{*}}\rightarrow 0$ and the compactness of $K' ,$
established in Lemma \ref{Preliminary}, imply that $(u_n)$ has a convergent
subsequence. $\hfill\Box$ 

\begin{lemma}[Compactness Condition - II]
\label{Compactness2}Assume that $(f)$, $(F_1)$, and $(F_5)$ hold. Then the
functional $I$ satisfies the compactness condition $(C)$, provided that $%
0<\mu <p.$
\end{lemma}

\paragraph{Proof.} Let $(u_n)$ be a sequence, in $D^{1,p}({\Bbb R}^N)$, 
satisfying 
\begin{equation}
\begin{array}{lll}
(i) &  & I(u_n)\rightarrow c \\ 
(ii) &  & (1+\| u_n\| _{D^{1,p}})\| I^{\prime
}(u_n)\| _{(D^{1,p})^{*}}\rightarrow 0.
\end{array}
\label{C3}
\end{equation}
Due to the same argument used in the proof of the previous lemma, we need
only to prove that the sequence $(u_n)$, is bounded. The proof will be
carried out by contradiction: suppose, up to a subsequence, that $\|
u_n\| _{D^{1,p}}\rightarrow \infty $. Proceeding as in the proof of
Lemma \ref{Compactness1}, condition $(F_1)$ together with (\ref{C3}) lead to 
\begin{equation}
\| u_n\| _{L_a^\mu }^{}\leq C.  \label{C4}
\end{equation}
Note that, by assumption $(F_1),$%
\[
\frac d{ds}\left( \frac{F(x,s)}{s^p}\right) =\frac{f(x,s)s-pF(x,s)}{%
s^{p+1}}\geq a(x)s^{\mu -p-1}, 
\]
for all $x\in {\Bbb R}^N$ and $s>0$. Now, integrating the above inequality
over an interval $[u,U]\subset (0,+\infty )$, we obtain 
\[
\frac{F(x,U)}{U^p}-\frac{F(x,u)}{u^p}\geq \frac{a(x)}{\mu -p}\left( \frac
1{U^{p-\mu }}-\frac 1{u^{p-\mu }}\right) . 
\]
From the hypothesis $\mu <p$, passing to the $\limsup$ in the last
inequality, as $U\rightarrow +\infty $, and using condition $(F_5)$,
 we obtain 
\[
F(x,u)\leq \frac 1p\omega (x)u^p-\frac{a(x)}{p-\mu }u^\mu ,\quad \forall
(x,u)\in {\Bbb R}^N\times (0,+\infty ). 
\]
Similarly, we show that 
\begin{equation}
F(x,u)\leq \frac 1p\omega (x)|u|^p-\frac{a(x)}{p-\mu }|u|
^\mu ,\quad \forall (x,u)\in {\Bbb R}^N\times {\Bbb R}.  \label{C5}
\end{equation}
Let $v_n=u_n/\| u_n\|_{D^{1,p}}$. By passing to
subsequences, we can find that $v_n$ converges weakly in $D^{1,p}({\Bbb R}%
^N) $ and almost everywhere to a function $v_0\in D^{1,p}({\Bbb R}^N).$
Therefore, 
\begin{equation}
\int\nolimits_{{\Bbb R}^N}\omega (x)|v_n|^p\,dx\rightarrow
\int\nolimits_{{\Bbb R}^N}\omega (x)|v_0|^p\,dx\,, \label{C6}
\end{equation}
because, by lemma \ref{Preliminary}, the functional $K(u)\doteq
\int\nolimits_{{\Bbb R}^N}\omega (x)|u|^pdx$ is weakly continuous.
Dividing (\ref{C3}.i) by $\| u_n\|_{D^{1,p}}^p$
and using estimate (\ref{C5}), we have 
\[
\frac 1p\int\nolimits_{{\Bbb R}^N}|\nabla v_n|^pdx-\frac
1p\int\nolimits_{{\Bbb R}^N}\omega (x)|v_n|^pdx+\frac{\|
u_n\| ^\mu }{(p-\mu )\| u_n\| _{D^{1,p}}}%
\leq \frac C{\| u_n\| _{D^{1,p}}^p}. 
\]
In view of $\| u_n\| _{D^{1,p}}\rightarrow \infty $, (\ref
{C4}), and (\ref{C6}), when passing to the limit we obtain
\[
1\leq \int\nolimits_{{\Bbb R}^N}\omega (x)|v_0|^p\,; 
\]
therefore, $v_0$ is not identically zero. On the other hand, from (\ref{C4}), we have 
\[
\int_{{\Bbb R}^N}a(x)|v_n|^\mu dx\leq \frac C{\|
u_n\| _{D^{1,p}}^\mu }. 
\]
Which by  Fatou's Lemma, implies
\[
\int_{{\Bbb R}^N}a(x)|v_0|^\mu dx\leq 0 
\]
and $v_0=0$, almost everywhere, which is a contradiction. 
$\hfill\Box$ 

\section*{Proof of Theorem 1}

To obtain a nontrivial critical point of the functional $I$,
will apply the following version of the Mountain-Pass Theorem, with
condition $(C)$ instead of the usual Palais-Smale compactness condition.

\begin{theorem}
Let $E$ be a real Banach space and $I\in C^1(E,{\Bbb R})$ satisfying condition $%
(C)$. Suppose that $I(0)=0$ and for some $\sigma ,\rho >0$ and $e\in E,$
with $\| e\| >\rho $, one has $\sigma \leq \inf_{\|
u\| =\rho }I(u)$ and $I(e)<0$. Then $I$ has a critical value $c\geq
\sigma $, characterized by 
\[
c=\inf_{\gamma \in \Gamma }\max_{0\leq \tau \leq 1}I(\gamma (\tau )), 
\]
where $\Gamma =\{\gamma \in C([0,1],E):\gamma (0)=0,\ \gamma (1)=e\}.$
\end{theorem}

\paragraph{Remark 4.}
It is not difficult to see that the same proof of the standard Mountain-Pass
Theorem (cf. \cite{Rabinowitz}) applies to the present
context; since the deformation theorem, Theorem 1.3 in \cite
{Bartolo-Benci-Fortunato}, is obtained with condition $(C)$ in a Banach
space framework. It is worth to observe also that this version of the
Mountain-Pass Theorem can be obtained as a consequence of Lemma 5 in \cite
{JMarcos}.

The proof of Theorem 1 follows from Lemma \ref{Compactness1}, where
the compactness condition is proved, and the next lemma, where the
geometric conditions are checked.

\begin{lemma}[Mountain-Pass Geometry]
\label{Geometry}Suppose $(f)$, $(F_3)$, and $(F_4)$ hold. Then there exist
positive constants $\sigma $ and $\rho $ such that $I(u)\geq \sigma $ if $%
\| u\| _{D^{1,p}}=\rho $. Moreover, there exists $\varphi \in
D^{1,p}({\Bbb R}^N)$ such that $I(t\varphi )\rightarrow -\infty $ as $%
t\rightarrow \infty .$
\end{lemma}

\paragraph{Proof.} Using the growth condition $(f)$ and $(F_3)$,
there exists a positive constant $C_\delta $ such that 
\[
F(x,u)\leq \frac{\alpha (x)|u|^p}p+C_\delta |u|^{p^{*}}. 
\]
Hence 
\begin{eqnarray*}
I(u) &\geq &\frac 1p\int_{{\Bbb R}^N}|\nabla u|^pdx-\frac 1p\int_{%
{\Bbb R}^N}\alpha (x)|u|^pdx-C_\delta \int_{{\Bbb R}^N}|u|
^{p^{*}}dx \\
&\geq &\frac 1p(1-\lambda _1^{-1}(\alpha ))\| u\| _{D^{1,p}}^p-%
\widehat{C}_\delta \| u\| _{D^{1,p}}^{p^{*}},
\end{eqnarray*}
for some positive constant $\widehat{C}_\delta $, where in the last
inequality we have used the variational characterization of the first
eigenvalue $\lambda _1(\alpha )$ and the Sobolev inequality. Since $\lambda
_1(\alpha )>1$, we can fix positive constants $\sigma $ and $\rho $ such
that $I(u)\geq \sigma $ if $\| u\| _{D^{1,p}}=\rho .$

Let us prove the second assertion. Consider $\varepsilon >0$ such that $%
\lambda _1(\omega )+\varepsilon <1$, and choose $\varphi \in C_0^\infty (%
{\Bbb R}^N)-\{0\}$ satisfying 
\[
\int_{{\Bbb R}^N}|\nabla \varphi |^pdx\leq (\lambda _1(\omega
)+\varepsilon )\int_{{\Bbb R}^N}\omega |\varphi |^pdx. 
\]
From $(F_4)$, we have 
\begin{eqnarray*}
\int_{{\Bbb R}^N}F(x,t\varphi )\,dx 
 &=&\int_{\{x:|t\varphi (x)|\geq R\}\cap \supp\varphi }F(x,t\varphi)\,dx\\
&+&\int_{\{x:|t\varphi(x)|\leq R\}\cap \supp\varphi }F(x,t\varphi )\,dx\\
&\geq &\frac{|t|^p}p\int_{\{x:|t\varphi (x)|\geq R\}\cap 
\supp\varphi }\omega (x)|\varphi |^p\,dx-C\,,
\end{eqnarray*}
for some positive constant $C$. $\,$Thus, 
\begin{eqnarray*}
I(t\varphi ) &\leq &\frac{|t|^p}p\left[ \int_{{\Bbb R}^N}|\nabla
\varphi |^p\,dx-\int_{\{x:|t\varphi (x)|\geq R\}\cap 
\supp\varphi }\omega |\varphi |^p\,dx\right] +C \\
&=&\frac{|t|^p}p\Big[ \int_{{\Bbb R}^N}|\nabla \varphi |
^pdx-\int_{{\Bbb R}^N}\omega |\varphi |^p\,dx\\
&+&\int_{\{x:|t\varphi
(x)|<R\}\cap \supp\varphi }\omega |\varphi |^pdx\Big]
+C\,.
\end{eqnarray*}
Since, by the Lebesgue convergence Theorem, 
\[
\lim_{|t|\rightarrow \infty }\int_{\{x:|t\varphi (x)|
<R\}\cap \supp\varphi }\omega |\varphi |^pdx=0, 
\]
there exists $M>0$, such that 
\[
\int_{\{x:|t\varphi (x)|<R\}\cap \supp\varphi }\omega |
\varphi |^pdx\leq -\frac{(\lambda _1(\omega )+\varepsilon -1)}2\int_{%
{\Bbb R}^N}\omega |\varphi |^pdx,\ \forall |t|\geq M. 
\]
This fact together with our choice of $\varphi $ imply that 
\[
I(tu)\leq \frac{|t|^p}p\frac{(\lambda _1(\omega )+\varepsilon -1)}%
2\int_{{\Bbb R}^N}\omega |\varphi |^pdx+C,\quad \forall |t|
\geq M. 
\]
Thus, we have $I(tu)<0$, for sufficiently large $|t|$. 
$\hfill\Box$

\section*{Proof of Theorem 2}

We shall obtain here the critical point by minimization. We have proved in
Lemma \ref{Compactness2} that assumptions $(F_1)$ and $(F_5)$ imply 
\[
F(x,u)\leq \frac 1p\omega (x)|u|^p-\frac{a(x)}{p-\mu }|u|
^\mu ,\quad \forall (x,u)\in {\Bbb R}^N\times {\Bbb R}. 
\]
Combining this estimate with the variational characterization of the first
eigenvalue $\lambda _1(\alpha )$, yields 
\begin{eqnarray*}
I(u) & \geq & \frac 1p\int_{{\Bbb R}^N}|\nabla u|^p\,dx-\frac
1p\int_{{\Bbb R}^N}\omega (x)|u|^p\,dx+\frac 1{p-\mu }\int_{{\Bbb R}%
^N}a(x)|u|^\mu\,dx \\ 
&\geq & \frac 1p(1-\lambda _1^{-1}(\omega ))\int_{{\Bbb R}^N}|\nabla
u|^p\,dx+\frac 1{p-\mu }\int_{{\Bbb R}^N}a(x)|u|^\mu\,dx.
\end{eqnarray*}
Thus, the functional $I$ is bounded below, since $\lambda _1(\omega )\geq 1.$
Therefore, in view of Lemma 3 in \cite{JMarcos} and Lemma \ref{Compactness2}%
, $I$ has a critical point $u$ which is its global minimum. 
$\hfill\Box$ 

\section*{Examples}

\paragraph{Example 1.}
Let $F:{\Bbb R}^N\times {\Bbb R\rightarrow R}$ given by 
\[
F(x,u)=\left\{ 
\begin{array}{lll}
\frac 1p\omega (x)|u|^p\ln |u|& if & u\neq 0, \\ 
0 & if & u=0,
\end{array}
\right. 
\]
where $\omega $ is a positive measurable function in $L^\infty ({\Bbb R}%
^N)\cap L^{N/p}({\Bbb R}^N)$ such that $\lambda _1(\omega )<1$. Notice that,
for all $x\in {\Bbb R}^N$ and $u\neq 0$, 
\[
u\frac{\partial F}{\partial u}(x,u)-\mu F(x,u)=\frac{(p-\mu )}p\omega
(x)|u|^p\ln |u|+\frac 1p\omega (x)|u|^p. 
\]
Thus, condition $(F_1)$ is satisfied with $\mu =p$, since 
\[
u\frac{\partial F}{\partial u}(x,u)-pF(x,u)=\frac 1p\omega \left( x\right)
|u|^p>0,\quad \forall x\in {\Bbb R}^N,\forall u\neq 0. 
\]
It is not difficult to see that the remaining hypotheses of Theorem 1 are
satisfied. On the other hand, if $\mu >p,$%
\[
u\frac{\partial F}{\partial u}(x,u)-\mu F(x,u)=[(p-\mu )\ln |u|
+1]\frac 1p\omega (x)|u|^p<0 
\]
for all $x$ $\in {\Bbb R}^N$ and $|u|$ sufficiently large.
Consequently, Ambrosetti-Rabinowitz condition $(AR)$ does not hold.

\paragraph{Example 2.}
Let $\psi :[1,+\infty )\rightarrow {\Bbb [}0,+\infty )$ be a continuous
nontrivial function satisfying $\int_1^{+\infty }\psi (t)dt<\infty $,
 and let 
$H(u)=d+\int_1^u[\psi (t)+t^{\mu -p-1}]\,dt$, $u\geq 1$, where $d$ is such
that $\lim_{u\rightarrow +\infty }H(u)=1$. 
Let $F(x,u)=\omega (x)u^pH(u)/p$ for $x\in {\Bbb R}^N,\ u\geq 1$, where $\omega $ is a positive measurable
function in $L^\infty ({\Bbb R}^N)\cap L^{N/p}({\Bbb R}^N)$, such that $%
\lambda _1(\omega )\geq 1$. Notice that $\lim_{u\rightarrow +\infty
}pF(x,u)/u^p=\omega (x)$ and that
\[
u\frac{\partial F}{\partial u}(x,u)-pF(x,u)=\frac 1p\omega
(x)u^{p+1}H' (u)\geq \frac 1p\omega (x)u^\mu ,\quad x\in {\Bbb R}%
^N,\ u\geq 1. 
\]
It is not difficult to see that $F$ can be extended as a function $%
F:{\Bbb R}^N\times {\Bbb R\rightarrow R}$ such that the conditions of
Theorem~2 are satisfied for all $u\in {\Bbb R}$ (cf. \cite{Costa-Magalhaes2}%
).

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\bigskip

{\sc Jo\~{a}o Marcos B. do \'{O}\newline
Departamento de Matem\'{a}tica, Universidade Federal da Para\'{\i}ba 
58059.900 Jo\~{a}o Pessoa, Pb Brazil}\newline
E-mail address: jmbo@terra.npd.ufpb.br

\end{document}