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\markboth{ Three solutions for quasilinear equations in near resonance }
{ Pablo De N\'apoli \& Mar\'{\i}a Cristina Mariani }
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 131--140.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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Three solutions for quasilinear equations in $\mathbb{R}^n$ near resonance
%
\thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J60.
 \hfil\break\indent
{\em Key words:} p-Laplacian, resonance, nonlinear eigenvalue problem.
 \hfil\break\indent
\copyright 2001 Southwest Texas State University.
\hfil\break\indent Published January 8, 2001. } }

\date{}
\author{ Pablo De N\'apoli \& Mar\'{\i}a Cristina Mariani }
\maketitle
\begin{abstract}
We use minimax methods to prove the existence of at least three solutions
for a quasilinear elliptic equation in $\mathbb {R}^n$ near
resonance.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lem}[theorem]{Lemma}
\newtheorem{rem}[theorem]{Remark}
\newtheorem{prop}[theorem]{Proposition}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12

\section{Introduction}

J. Mawhin and K. Smichtt \cite{MS}, proved the existence of at
least three solutions for the two-point boundary value problem
\[\displaylines{
-u''-u+\varepsilon u=f(x,u)+h(x)\cr
 u(0)=u(\pi)=0
}\]
 for \( \varepsilon >0 \) small enough, \( h \) orthogonal to \( \sin  x \)
and \( f \) bounded satisfying the sign condition \( uf(x,u)>0 \).
In \cite{TS}, To Fu Ma and L. Sanchez considered the problem
\begin{equation}\label{problema-1}
-\Delta _{p}u-\lambda_{1}|u|^{p-2}u+\varepsilon
|u|^{p-2}u=f(x,u)+h(x)
\end{equation}
 in \(W_0^{1,p}(\Omega)\) with \( \Omega \subset \mathbb {R}^n \)
 a bounded domain,  and \( \lambda _{1} \)
the first eigenvalue of
\begin{eqnarray}\label{problema-2}
& -\Delta _{p}u=\lambda |u|^{p-2}u\quad \hbox{in }\Omega &\\
& u=0\quad\hbox {on } \partial \Omega \,.&\nonumber
\end{eqnarray}
They proved the following result.

\begin{theorem}
Suppose that \( p\geq 2 \) and that the following two conditions
hold:\begin{enumerate}

\item[(H1)] \( f:\overline{\Omega }\times \mathbb {R}^n\to
\mathbb {R}^n \) is a continuous function and there exist \(
\theta >\frac{1}{p} \) such that
$\theta sf(x,s)-F(x,s)\to -\infty$ as $|s|\to \infty$
\item[(H2)] There exists \( R>0 \) such that
$sf(x,s)>0$ for all $x\in \Omega$, $|s|\geq R$
\end{enumerate}
 Then for every \( h\in L^{p'}(\Omega ) \)
 with \( \int _{\Omega }h(x)\varphi _{1}(x)dx=0 \),
where \( \varphi _{1} \) is the first eigenfunction of
(\ref{problema-2}), the equation (\ref{problema-1}) has at least
three solutions for \( \varepsilon >0 \) small enough.
\end{theorem}

We recall that the assumptions on \( f \) imply the growth
condition
\[
|f(x,s)|\leq c_{1}+c_{2}|s|^{\sigma }\]
 with \( \sigma =\frac{1}{\theta }<p \).

These  problems have been studied for several authors, see
 \cite{DJM,FMST,G,O}.

\subsubsection*{The functional setting}

Our aim is to extend this result to equations in \( \mathbb
{R}^n \). As \( W^{1,p}(\mathbb {R}^n) \) is no longer
compactly imbedded into \( L^{p}(\mathbb {R}^n) \), we shall
work in the space \( D^{1,p} \), the closure of \(
C^{1}_{0}(\mathbb {R}^n) \) with the norm
\[
\left\| u\right\| _{1,p}=\Big( \int _{\mathbb {R}^n}|\nabla
u(x)|^{p}dx\Big) ^{1/p}\]
 By the Sobolev inequality we have:
$D^{1,p}\subset L^{p^{*}}(\mathbb {R}^n)$
 with \( p^{*}=\frac{Np}{N-p} \), this imbedding is not compact,
 however in proposition
\ref{prop-compacidad} we prove that the imbedding \(
D^{1,p}\subset L^{p}_{g}(\mathbb {R}^n) \) is compact for \(
g\in L^{N/p}\cap L_{loc}^{N/p+\varepsilon } \).

\subsubsection*{Simplicity of the first eigenvalue}

We recall the simplicity of the first eigenvalue of the
p-laplacian that is proved in \cite{FMST}. They studied the
problem:
\begin{eqnarray}\label{eigenvalue-problem}
&-\Delta _{p}u=g(x)|u|^{p-2}u\quad x\in \mathbb {R}^n \\
&0<u \quad \hbox {in} \; \mathbb {R}^n,\quad
\lim _{|x|\to +\infty}u(x)=0\,,\nonumber
\end{eqnarray}
where \( 1<p<n \). They proved the theorem below, assuming the
following conditions:

\par $(G)$ $g$ is a smooth function, at least \( C_{loc}^{0,\gamma
}(\mathbb {R}^n) \) for some \( \gamma \in (0,1) \), such that
\( g\in L^{N/p}(\mathbb {R}^n)\cap L^{\infty }(\mathbb {R}^n)
\) and \( g(x)>0 \) in \( \Omega ^{+} \) with \( \left| \Omega
^{+}\right| >0 \). Also $g$ satisfies one the following two conditions
\begin{description}
\item{$(G^{+})$} \( g(x)\geq 0 \) a.e. in \( \mathbb {R}^n \)
\item{$(G^{-})$} \( g(x)<0 \) for \( x\in \Omega ^{-} \),
with \( |\Omega ^{-}|>0 \).
\end{description}

\begin{theorem}
\begin{enumerate}
\item Let \( g \) satisfy $(G)$ and \( (G^{+}) \). Then equation (\ref{eigenvalue-problem})
admits a positive first eigenvalue,
\begin{equation}
\label{minimization-problem} \lambda _{1}=\inf _{B(u)=1}\left\|
u\right\| _{D^{1,p}}^{p}
\end{equation}
 with $B(u)=\int _{\mathbb{R}^n }|u(x)|^{p}g(x)\,dx$.

\item Let \( g \) satisfy $(G)$ and \( (G^{-}) \). Then problem (\ref{eigenvalue-problem})
admits two first eigenvalues of opposite sign:
\[
\lambda ^{+}_{1}=\inf _{B(u)=1}\left\| u\right\|_{D^{1,p}}^{p}\quad
\lambda ^{-}_{1}=-\inf _{B(u)=-1}\left\| u\right\|_{D^{1,p}}^{p}\]
In both cases the associated eigenfunctions \( \varphi ^{+}_{1}
\), \( \varphi ^{-}_{1} \) belong to \( D^{1,p}\cap L^{\infty }
\).

\item The set of eigenvectors corresponding to \( \lambda _{1} \) is a one dimensional
subspace.
\end{enumerate}\end{theorem}

\begin{rem}
The first eigenfunction \( \varphi _{1} \) does not change its
sign in \( \Omega  \), so we may assume \( \varphi _{1}\geq 0 \).
\end{rem}
\paragraph{Proof.}
Taking \( \varphi ^{-} \) as a test function in
(\ref{eigenvalue-problem}) with \( \lambda =\lambda _{1} \) we see
that
\[
\int _{\mathbb{R}^n }|\nabla (\varphi ^{-})|^{p}=\lambda _{1}\int
_{\mathbb{R}^n }|\varphi ^{-}_{1}|^{p}g(x)dx\]

It follows that \( \varphi ^{-}=0 \) (and \( \varphi \geq 0 \) ),
or \( \varphi ^{-}_{1} \) is also a solution of the minimization
problem (\ref{minimization-problem}). In the last case, from the
simplicity of the first eigenvalue \( \varphi _{1}^{-}=c\varphi
_{1} \). It follows that \( \varphi ^{-}=-\varphi _{1} \), so \(
\varphi _{1}\leq 0 \). \hfill$\diamondsuit$

\subsection*{Existence of multiple solutions}

In this paper we study quasilinear elliptic
equation
\begin{equation}
\label{our-problem} -\Delta _{p}u=(\lambda _{1}-\varepsilon
)g(x)|u|^{p-2}u+f(x,u)+h(x)
\end{equation}
 in \( \mathbb {R}^n \). We assume the following:

\begin{enumerate}
\item \( 1<p<n \) and \( \varepsilon >0 \)
\item On the weight \( g \) we make the assumptions \( (G) \) and \( (G^{+}) \) of \cite{FMST}
\item \( h\in L^{p^{*\prime }} \) and \( \int _{\mathbb {R}^n}h\varphi _{1}dx=0 \)
\item We assume that the non linearity \( f:\mathbb {R}^n\times \mathbb {R}\to \mathbb {R} \)
is continuous and satisfies

\begin{description}
\item{(H0)} Growth condition.
\[
|f(x,s)|\leq c_{1}(x)+c_{2}(x)|s|^{\sigma -1}\]
 with \( \sigma <p \) and \( c_{1}\in L^{(p^{*})'} \),
 \( c_{2}\in L^{(p^{*}/\sigma )'}\cap L_{loc}^{(p/\sigma )'+\eta } \)
for some \( \eta >0 \).
\item{(H1)} If \( F(x,s)=\int ^{s}_{0}f(x,t)dt \) then
$\frac{1}{p}sf(x,s)-F(x,s)\to -\infty$ as $|s|\to \infty$.
\item{(H2)} Sign condition. There exists \( R>0 \) such that:
$sf(x,s)>0$ for all $x\in \mathbb {R}^n$, $|s|\geq R$.
\end{description}
\end{enumerate}

\par For example we may take $f(x,s)= c_2(x) |s|^{\sigma-1}s \cdot \rm{sgn}\;s$ where $c_2(x)$
satisfies the conditions above, $c_2(x)>0$, and $\sigma<p$.

\par Note that integrating on condition \( (H0) \) we get
\[
F(x,s)\leq c_{1}(x)|s|+c_{2}(x)\frac{|s|^{\sigma }}{\sigma }\,.
\]
 In the next section we will see that for the functional
\( C(u)=\int _{\mathbb{R}^n} F(x,u)dx \) to be of class \(
C^{1}(D^{1,p}(\mathbb {R}^n)) \), condition \( (H0) \) is the
natural choice.

Our main result is the following theorem:

\begin{theorem}
Under the assumptions above, problem (\ref{our-problem}) has at
least three solutions for \( \varepsilon >0 \) small enough.
\label{our-main}
\end{theorem}

\section{Technical Lemmas}

For the proof of theorem \ref{our-main} we will need the following
results:

\subsection*{A compactness result in weighted \protect\protect\protect\( L^{p}\protect \protect \protect \)
spaces}

If \( u\in D^{1,p} \), \( 1 \leq q \leq p^* \), \(
\frac{1}{r}+\frac{q}{p^{*}}=1 \; \)and \( g\in L^{r},g\geq 0 \),
then from H\"older and Sobolev inequalities, we have that
\begin{equation}
\label{imbedding} \int _{\mathbb {R}^n}|u|^{q}g\leq C\int
_{\mathbb {R}^n}|\nabla u|^{p}
\end{equation}
 and it follows that \( D^{1,p}\subset L_{g}^{q} \). The following result
proves that under appropriate conditions, this imbedding is also
compact. (Other previous results can be found in \cite{KP}).

\begin{prop}
Let \( 1\leq q < p^{*} \), \( \frac{1}{r}+\frac{q}{p^{*}}=1 \), \(
g\in L^{r}\cap L_{loc}^{r+\varepsilon } \) for some \( \varepsilon
>0 \). Then the imbedding
\[
D^{1,p}\subset L_{g}^{q}(\mathbb {R}^n)\]
 is compact.\label{prop-compacidad}
\end{prop}
\paragraph{Proof.}
Let \( (u_{n})\subset D^{1,p} \) be a bounded sequence:
\[
\left\| u_{n}\right\| _{1,p}\leq C\]
Then, as \( D^{1,p} \) is reflexive, we may extract a weakly
convergent subsequence \( (u_{n_{k}}) \). For simplicity we assume
that \( u_{n}\rightharpoonup u \). We want to prove that in fact
\( u_{n}\to u \) strongly. From H\"older and Sobolev
inequalities we have:
\[
\int _{|x|>R}g|u-u_{n}|^{q}\leq \Big( \int _{|x|>R}|g|^{r}\Big)
^{1/r}\Big( \int _{|x|>R}|u_{n}-u|^{p^{*}}\Big) ^{p/p^{*}}\leq
C\Big( \int _{|x|>R}|g|^{r}\Big) ^{1/r}\]
 Given \( \varepsilon >0 \), as \( g\in L^{r} \) we can choose \( R>0 \)
verifying
\[
\int _{|x|>R}g|u-u_{n}|^{q}\leq \frac{\varepsilon }{2}\]
 Now \( D^{1,p}(\mathbb {R}^n)\subset W_{loc}^{1,p}(\mathbb {R}^n) \) continously
and by the Rellich-Kondrachov theorem
\[
u_{n}\to u \;\hbox {strongly} \;\hbox {in} \;
L^{t}(B_{R})\]
 if \(1 \leq t<p^{*} \). We choose \( s>1 \) such that \( s'=r+\varepsilon
 \), then \( s<\frac{p^{*}}{q} \), and
\[
\int _{|x|\leq R}g|u_{n}-u|^{q}\leq \Big( \int _{|x|\leq
R}|g|^{s'}\Big) ^{1/s'}\Big( \int
_{|x|<R}|u-u_{n}|^{qs}\Big) ^{1/s}\leq \frac{\varepsilon }{2}\]
 if \( n\geq n_{0}(\varepsilon ) \). So \( u_{n}\to u \) in \( L^{p}_{g}(\mathbb {R}^n) \).
\hfill$\diamondsuit$

\subsection*{Some results about the Associated Functional}

Under the same assumptions of theorem \ref{our-main}, we have the
following results:

\begin{lem}
Let \( C:D^{1,p}(\mathbb {R}^n)\to \mathbb {R} \) given by \(
C(u)=\int _{\mathbb{R}^n} F(x,u)dx \). Then \( C\in
C^{1}(D^{1,p}(\mathbb{R}^n )) \) and \( C'(u)(h)=\int
_{\mathbb{R}^n }f(x,u)h \)
\end{lem}
\paragraph{Proof.}
From the H\"older inequality we have that
\[
|C(u)|\leq \int _{\mathbb{R}    ^n}c_{1}(x)|u|+c_{2}(x)\frac{|u|^{\sigma
}}{\sigma }dx\leq \left\| c_{1}\right\| _{(p^{*})'}
\left\| u\right\| _{p^{*}}+\frac 1{\sigma} \left\|
c_{2}\right\| _{(p^{*}/\sigma )'}\left\|
u\right\| ^{\sigma }_{p^{*}}\]
 From the imbedding \( D^{1,p}\subset L^{p^{*}} \) we conclude that \( C(u) \)
is well defined. In a similar way,
\begin{eqnarray*}
\big| \int _{\mathbb{R}^n }f(x,u)h\big|
&\leq& \int _{\mathbb{R}^n}c_{1}(x)|h|+c_{2}|u|^{\sigma -1}|h|\\
&\leq& \left\|c_{1}\right\| _{(p^{*})'}\left\| h\right\|
_{p^{*}}+\left\| c_{2}\right\| _{(p^{*}/\sigma )'}
\left\| u\right\| _{p^{*}}^{\sigma -1}\left\|
h\right\| _{p^{*}}
\end{eqnarray*}
 and we have that \( \int _{\mathbb{R}^n }f(x,u)h \) is also well defined.
Using a similar argument as in \cite{O}, we conclude the proof.
\hfill$\diamondsuit$
\begin{lem}
Assume that $f(x,y)$ is a Caratheodory function, verifying that
\[
|f(x,u)|\leq c_{1}(x)+c_{2}(x)|u|^{\sigma -1}\]
 where \( 1\leq \sigma <p^{*} \), \( c_{1}\in L^{s_{1}}(\mathbb {R}^n) \)
with \( s_{1}={p^*}^{\prime} \), and \( c_{2}\in L^{s_{2}}\cap
L_{loc}^{s_{2}+\varepsilon } \) with \(
s_{2}=\frac{p^{*}}{p^{*}-\sigma } \).
Then the Nemitski operator \( N_{f}:D^{1,p}(\mathbb
{R}^n)\to L^{p^{{*}'}}(\mathbb {R}^n) \)
given by \( N_{f}(u)=f(x,u) \) is compact. \label{n}
\end{lem}

\paragraph{Proof.}
Let \( (u_{n}) \) be a sequence in \( D^{1,p} \) such that \(
u_{n}\rightharpoonup u \) weakly in \( D^{1,p} \). We may assume,
passing to a subsequence, that $u_{n}\to u \; \hbox{a.e.}$.

As \( \sigma <p^{*} \), we apply proposition \ref{prop-compacidad}
with \( q=(\sigma -1)s_{1}<p^{*} \),  \( g=c^{s_{1}}_{2} \). We
note that \( g\in L^{r}\cap L_{loc}^{r+\varepsilon^\prime } \)
with \( r=\frac{p^{*}-1}{p^{*}-\sigma } \). We get, passing to a
subsequence, that
$u_{n}\to u$ in  $L_{g}^{q}$.

 From theorem IV.9 in \cite{B}, we obtain, after passing again to a
subsequence, a function \( m\in L_{g}^{q}(\mathbb {R}^n) \) such
that
\[
|u_{n}(x)|\leq m(x)\]
 \( a.e. \) with respect to the measure \( g(x)dx \). Then, from condition
 \( (H0) \) we deduce that
\begin{eqnarray*}
|f(x,u)-f(x,u_{n})|^{s_{1}}&\leq&
2^{s_{1}}[|f(x,u)|^{s_{1}}+|f(x,u_{n})|^{s_{2}}]\\
&\leq& 2^{s_1+1}[c_{1}(x)^{s_{1}}+c_{2}(x)^{s_{1}}|m|^{(\sigma
-1)s_{1}}]\,.
\end{eqnarray*}
Applying the bounded convergence theorem to \( \int _{\mathbb{R}^n
}|f(x,u)-f(x,u_{n})|^{s_1}dx \) we obtain that \( f(x,u_{n})\to
f(x,u) \) in \( L^{s_{1}}(\mathbb {R}^n) \). \hfill$\diamondsuit$

\begin {rem}
The weak solutions of equation (\ref{our-problem}) are the
critical points in \( D_{0}^{1,p} \) of the functional
\[
J_{\epsilon }(u)=\frac{1}{p}\int_{\mathbb{R}^n} |\nabla
u|^{p}dx-\frac{\lambda _{1}-\varepsilon }{p}\int_{\mathbb{R}^n
}|u|^{p}g(x)dx-\int_{\mathbb{R}^n} (F(x,u)+h(x)u)\]
 Under the previous assumptions it is easy to check that \( J_{\varepsilon }\in C^{1}(D^{1,p}) \).
\end {rem}

Let
\[
W=\left\{ w\in D^{1,p}:\int_{\mathbb{R}^n} g(x)|\varphi _{1}|^{p-2}\varphi
_{1}w=0\right\} \]
 We recall that as a consequence of proposition \ref{prop-compacidad} \( W \)
is a weakly closed linear subspace.

\begin{lem} \label{lema1}
If \( \varepsilon <\lambda _{1} \) , \( J_{\epsilon } \) is
coercive in \( D^{1,p} \), and there exist \( m>0 \) such that
$\inf_{u\in W} J_{\epsilon }(u)\geq -m$.
\end{lem}
\paragraph{Proof.}
We suppose \( 0<\varepsilon <\lambda _{1} \), then
\[
J_{\varepsilon }(u)\geq \frac{1}{p}\left( 1-\frac{\lambda
_{1}-\epsilon }{\lambda _{1}}\right)
 \int _{\mathbb{R}^n }|\nabla u|^{p}-\int _{\mathbb{R}^n }(F(x,u)+hu)\]
and
\[
J_{\varepsilon }(u)\geq \frac{\epsilon }{p\lambda _{1}}\left\|
u\right\| ^{p}_{1,p}-C_{1}-C_{2}\left\| u\right\|
_{1,p}^{\sigma }-\left\| h\right\| _{(p^{*})^{\prime
}}\left\| u\right\| _{p^{*}}\]
 As \( \sigma <p \), it follows that \( J_{\varepsilon } \) is coercive.

We define
\[
\lambda _{W}=\inf \left\{ \int _{\mathbb {R}^n}|\nabla
w|^{2}:w\in W,\int _{\mathbb {R}^n}g(x)|w(x)|^{p}=1\right\} \]
 We claim that \( \lambda _{W}>\lambda _{1} \). In fact if \( \lambda _{1}=\lambda _{W} \)
then we would have \( w\in W \) verifying
\[
\int _{\mathbb{R}^n }|w|^{p}=\lambda _{1},\int _{\mathbb{R}^n
}|w|^{p}g(x)dx=1\]
 So by the simplicity of the first eigenvalue, \( w=c\varphi _{1} \) but this
contradicts the definition of \( W \).

Then, for \( u\in W \) we have
\[
J_{\varepsilon }(u)\geq \frac{\lambda _{W}-\lambda _{1}}{p\lambda
_{W}}\left\| u\right\| ^{p}_{1,p}-C_{1}-C_{2}\left\|
u\right\| _{1,p}^{\sigma }-\left\| h\right\|
_{(p^{*})'}\left\| u\right\| _{p^{*}}\]
 Then \( J_{\varepsilon } \) is uniformly coercive in \( W \) respect to \( \varepsilon  \),
and in particular is uniformly bounded from below.
\hfill$\diamondsuit$

For stating the next result we need the two open sets:
\[\displaylines{
O^{+}=\Big\{ w\in D^{1,p}:\int_{\mathbb{R}^n}  g(x)|\varphi
_{1}|^{p-2}\varphi _{1}w>0\Big\}, \cr O^{-}=\Big\{ w\in
D^{1,p}:\int_{\mathbb{R}^n}  g(x)|\varphi _{1}|^{p-2}\varphi
_{1}w<0\Big\} } \]

The next condition is a variant of the Palais-Smale condition
(PS).

We will say that a functional \( \phi:D^{1,p} \to \mathbb{R} \)
verifies the \((PS)_{O^{\pm },c}\) condition if any sequence \(
(u_{n}) \) in \( O^{+} \) (respectively in \( O^{-} \)) with \(
\phi (u_{n})\to c \), \( \phi' (u_{n})\to 0 \), has a subsequence
\( (u_{n_{k}})\to u\in O^{+} \).

\begin{prop}
The operator \( -\Delta _{p}:D^{1,p}\to (D^{1,p})^{*} \)
satisfies the \( (S_{+}) \) condition: if \( u_{n}\rightharpoonup u
\) (weakly in \( D^{1,p}(\mathbb {R}^n) \) ) and \( \lim \sup
_{n\to \infty } \left\langle -\Delta
_{p}u_{n},u_{n}-u\right\rangle \leq 0 \), then \( u_{n}\to
u \) (strongly in \( D^{1,p} \) )
\end{prop}
\paragraph{Proof.}
This follows from the uniform convexity of \( D^{1,p}(\mathbb
{R}^n) \) (see \cite{DJM})

\begin{lem}
\( J_{\epsilon } \) satisfies the \( (PS) \) condition, and it
verifies \( (PS)_{O^{\pm },c} \) if \( c<-m \).\label{lema2}
\end{lem}
\paragraph{Proof.}
Let \( (u_{n})\subset D^{1,p} \) be a \( (PS) \) sequence such that
\[
J_{\varepsilon }(u_{n})\to c, J_{\varepsilon }^{\prime
}(u_{n})\to 0\]
 Since \( J_{\varepsilon }\; \) is coercive, it follows that \( (u_{n}) \) is
bounded in \( D^{1,p} \), which is reflexive, so (after passing to
a subsequence) we may assume that \( u_{n}\to u \) weakly.
We want to show that in fact, \( u_{n}\to u \) strongly.
We have that
\begin{eqnarray*}
J_{\varepsilon }'(u_{n})(u_{n}-u)
&=&\int |\nabla u_{n}|^{p-2}\nabla u_{n}\cdot \nabla (u_{n}-u)\\
&&-(\lambda_{1}-\varepsilon )\int |u_{n}|^{p-2}u_{n}(u_{n}-u)g(x)dx \\
&&-\int h(u_{n}-u)-\int f(x,u_{n})(u_{n}-u)
\end{eqnarray*}
 Clearly \( \int h(u_{n}-u)\to 0 \) since \( u_{n}\rightharpoonup u \)
weakly. Then
\( u_{n}\to u \) strongly in \( L_{g}^{p}(\mathbb {R}^n)
\) since the imbedding \( D^{1,p}\subset L_{g}^{p} \) is compact.
It follows that: \( \int
|u_{n}|^{p-2}u_{n}(u_{n}-u)g(x)dx\to 0 \)

From proposition \ref{n} and the H\"older inequality
\[
\int f(x,u_{n})(u_{n}-u)dx = \int [f(x,u)-f(x,u_n)](u_n-u) dx +
\int f(x,u)(u_n-u) \to 0\,.\]
Since \(J_{\varepsilon }'(u_{n})(u_{n}-u)\to 0\), it
follows that
\[
\int |\nabla u_{n}|^{p-2}\nabla u_{n}\cdot \nabla
(u_{n}-u)dx\to 0\]
 or equivalently,
$\left\langle -\Delta _{p}u_{n},u_{n}-u\right\rangle \to 0$.
 By the \( S_{+} \) condition, this implies that \( u_{n}\to u \)
strongly in \( D^{1,p} \).

To prove that \( J_{\epsilon } \) satisfies \(
(PS)_{O^{\pm },c} \) for \( c<-m \), consider \( (u_{n})\subset
O^{\pm } \) be a \( (PS)_{c} \) sequence. There exists a
convergent subsequence: \( u_{n_{k}}\to u \), and it is
enough to prove that \( u\in O^{\pm } \), but if \( u\in \partial
O^{\pm }=W \), then \( c=J(u)\geq -m \), a contradiction.
\hfill$\diamondsuit$
\begin{lem}
If \( \varepsilon >0 \) is small enough, there exists two numbers,
$t^{-}<0<t^{+}$,
 such that \( J_{\varepsilon }(t^{\pm }\varphi _{1})<-m \).\label{lema3}
\end{lem}
\paragraph{Proof.}
From \( \int h(x)\varphi _{1}(x)dx=0 \), we have that
\[
J_{\varepsilon }(t\varphi _{1})=\frac{1}{p}\int_{\mathbb{R}^n}94
\varepsilon t^{p}\varphi ^{p}_{1}g(x)-\int_{\mathbb{R}^n}
F(x,t\varphi _{1}(x))dx\,.\] Since \( \varphi _{1}\in L^{\infty }
\), we can assume that $0\leq \varphi _{1}(x)\leq 1$ for all $x\in
\mathbb {R}^n$.

 First, since \( \varphi ^{p}_{1}g\in L^{1} \), we can choose \( \rho  \)
big enough, such that:
\[
\frac{1}{p}\int _{|x|>\rho }\varphi ^{p}_{1}gdx<\frac{m}{2}\]
 and we split the integral \( J_{\varepsilon } \) in two parts:
$J_{\varepsilon }=J_{\varepsilon }^{1}+J_{\varepsilon }^{2}$,
 where \( J_{\varepsilon }^{1} \) is the integral over \( |x|\leq \rho  \),
and \( J_{\varepsilon }^{2} \) is the integral over \( |x|>\rho
\).

We define
\[\displaylines{
A(t)=\{x:|x|\leq \rho :\varphi _{1}(x)>R/t\}\cr
B(t)=\{x:|x|\leq \rho :\varphi _{1}(x)\leq R/t\}
}\]
 Then
\[
\int _{B(t)}[\frac{\varepsilon }{p}t^{p}\varphi
^{p}_{1}-F(x,t\varphi _{1}(x))]dx\]
 is uniformly bounded in \( \varepsilon  \) and \( t \) for \( \varepsilon \leq \varepsilon _{0} \).
Let
\begin{eqnarray*}
M_{\varepsilon }(t)&=&\int _{A(t)} \left( \frac{1}{p}t\varphi
_{1}(x)f(x,t\varphi _{1}(x))-F(x,t\varphi _{1}(x)) \right) \\
&&+\int_{B(t)} \left[\frac{\varepsilon }{p}t^{p}\varphi
^{p}_{1}-F(x,t\varphi _{1}(x))\right]dx
\end{eqnarray*}
Then, from \( (H1) \) and Fatou lemma,
$M_{\varepsilon }(t)<-2m$  for \( t \) big enough, and
\( \varepsilon \leq \varepsilon _{0} \).

By \( (H2) \) there exists \( 0<\varepsilon _{t}\leq \varepsilon
_{0} \) such that
\[
\varepsilon _{t}u^{p-1}g(x)<f(x,u) \;\hbox {in}\;
\overline{B_{\rho }}\times [R,t]\]
 Then if \( \varphi _{1}(x)>R/t \) and \( |x|\leq \rho  \) we have:
\[
\varepsilon _{t}t^{p-1}\varphi _{1}(x)^{p-1}g(x)<f(x,t\varphi
_{1})\]
and
\[
J_{\varepsilon }^{1}(t\varphi _{1}) \leq M_\varepsilon(t)<-2m\,.\]
From \( (H2) \), since \( F(x,t\varphi _{1})\geq 0 \), if we
choose \( \varepsilon _{t} \) satisfying
$\varepsilon _{t}<\frac{1}{t^{p}}$ then,
\[
J_{\varepsilon _{t}}^{2}(t\varphi _{1})\leq \frac{1}{p}\int
_{|x|>\rho }\varepsilon _{t}t^{p}\varphi ^{p}_{1}dx<\frac{m}{2}\]
 and we conclude that
$J_{\varepsilon _{t}}(t\varphi _{1})<-m$
for any \( \varepsilon _{t}\leq \varepsilon _{0} \). In a similar way,
choosing first \( t \) big enough, and then \( \varepsilon _{t} \) small,
we can prove that \( J_{\varepsilon _{t}}(-t\varphi _{1})<-m \)
\hfill$\diamondsuit$

\subsection*{Proof of theorem \ref{our-main}}

For \( \varepsilon >0 \) small enough, from lemmas \ref{lema2} and
\ref{lema3} we have that
\[
-\infty <\inf _{O^{\pm }}J_{\varepsilon }<-m\]
 and since \( (PS)_{c,O^{\pm }} \) holds for all \( c<-m \), it follows from
the deformation lemma that the above infima are attained, say at
\( u^{-}\in O^{-} \) and \( u^{+}\in O^{+} \). Since \( O^{\pm }
\) are both open in \( D^{1,p} \) we have found two critical
points of \( J_{\varepsilon } \).
Let
\[
c=\inf _{\gamma \in \Gamma }\max _{t\in [0,1]}J_{\varepsilon
}(\gamma (t))\]
 with
\[
\Gamma =\{\gamma \in C([0,1],D^{1,p}(\mathbb {R}^n):\gamma
(0)=u^{-},\gamma (1)=u^{+}\}\]
We observe that \( \gamma ([0,1])\cap W\neq 0 \) for any \( \gamma
\in \Gamma  \), so we conclude that
\[
c=\inf _{W} J_{\varepsilon }\geq -m\]
 \( J_{\varepsilon } \) verifies \( (PS) \), and from Ambrossetti-Rabinowitz's
Mountain Pass Theorem \cite{AR} we conclude that \( c \) is a
third critical value of \( J_{\varepsilon } \), and since \(
J_{\varepsilon }(u^{\pm })<-m \), the corresponding critical point
is different from \( u^{+},u^{-} \).

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\noindent{\sc Pablo L. De N\'apoli} (e-mail: pdenapo@dm.uba.ar) \\
{\sc M. Cristina Mariani} (e-mail: mcmarian@dm.uba.ar) \\[2pt]
Universidad de Buenos Aires \\
FCEyN - Departamento de Matem\'atica \\
Ciudad Universitaria, Pabell\'on I \\
Buenos Aires, Argentina
 \end{document}