\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 $$\label{problema-1} -\Delta _{p}u-\lambda_{1}|u|^{p-2}u+\varepsilon |u|^{p-2}u=f(x,u)+h(x)$$ 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, $$\label{minimization-problem} \lambda _{1}=\inf _{B(u)=1}\left\| u\right\| _{D^{1,p}}^{p}$$ 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 $$\label{our-problem} -\Delta _{p}u=(\lambda _{1}-\varepsilon )g(x)|u|^{p-2}u+f(x,u)+h(x)$$ in $$\mathbb {R}^n$$. We assume the following: \begin{enumerate} \item $$10$$ \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 $\sigma0 \) 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 $$\label{imbedding} \int _{\mathbb {R}^n}|u|^{q}g\leq C\int _{\mathbb {R}^n}|\nabla u|^{p}$$ 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 $$1 \leq t1$$ 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|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 $$\sigma \lambda _{1}$$. 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\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)R/t \) and $$|x|\leq \rho$$ we have: \[ \varepsilon _{t}t^{p-1}\varphi _{1}(x)^{p-1}g(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 $$u^{+},u^{-}$$. \begin{thebibliography}{00} \bibitem{AR} A. Ambrosetti \& P. H. Rabinowitz, {\em Dual Variational Methods in Critical Point Theory and Applications}, Jounal Functional Analysis 14 (1973) pp.349-381 \bibitem{B} H. Brezis, Analyse fonctionnelle, Masson, Paris 1983 \bibitem{DJM} G. Dinca, P. Jebelean, \& J. Mawhin, {\em Variational and Topological Methods for Dirichlet Problems with p-Laplacian}- Recherches de math‰matique (1998) Inst. de Math. Pure et. Apliqu‰e, Univ. Cath. de Louvain. \bibitem{FMST} J. Fleckinger, R.F. Man\'asevich, N.M. Stavrakakis, \& F. De Thëlin, {\em Principal Eigenvalues for Some Quasilinear Elliptic Equations on $$\mathbb {R}^n$$}, Advances in Differential Equations - Vol. 2, Number 6, November 1997 ,pp. 981-1003 \bibitem{G} J.-P. Gossez, {\em Some Remarks on the Antimaximum Principle}, Revista de la Uni\'{o}n Matem\'{a}tica Argentina- vol. 41, 1 (1998) pp. 79-84 \bibitem{KP} I. Kuziw \& S. Pohozaev, {\em Entire Solutions of Semilinear Elliptic Equations}, Progress in Nonlinear Differential Equations and Their Applications - Vol 33. - Birkhauser \bibitem{MS} Mawhin J. \& Schmitt K., {\em Nonlinear eigenvalue problems with the parameter near resonance}, Ann. Polonici Math. LI (1990) pp. 241-248 \bibitem{O} Jo\~ao Marcos B. do \'O, {\em Solutions to perturbed eigenvalue problems of the p-Laplacian in $$\mathbb {R}^n$$}, Electronic Journal of Diff. Eqns., Vol. 1997(1997), No. 1, pp. 1-15 \bibitem{TS} To Fu Ma \& L. Sanchez, {\em Three solutions of a Quasilinear Elliptic Problem Near Resonance}, Universidade de Lisboa CAUL/CAMAF-21/95 \end{thebibliography} \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}