\documentclass[twoside]{article} \usepackage{amsfonts} % used for R in Real numbers \pagestyle{myheadings} \setcounter{page}{131} \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)} \vspace{\bigskipamount} \\ % 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 \( \sigma =\frac{1}{\theta }
0 \) 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 \( 1
0 \) \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 \( \sigma
0 \). \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
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 \(1 \leq t
1 \) 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| \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