\documentstyle[twoside, amssymb]{article} %\input amssym.def % used for R in Real numbers \pagestyle{myheadings} \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 $$-\Delta _pu\equiv -\diver\left( |\nabla u| ^{p-2}\nabla u\right) =f(x,u)\quad\mbox{in}\quad {\Bbb R}^N\,, \label{1}$$ 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 s0,\quad \forall (x,u)\in {\Bbb R}% ^N\times ({\Bbb R-\{}0\}), \] \item{$(F_2)$} For some$p0, \] 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 $0p$. \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 $% {\Bbb R}^N$, $00$, we choose $r_n$ sufficiently large such that $$\max \{\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))},\| b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}\}<\epsilon /2C\,. \label{D2}$$ 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 $$|K_n(u_k)-K_n(u)|<\epsilon /2\,. \label{D3}$$ 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 $$\begin{array}{lll} (i) & & J(u_n)\rightarrow c \\ (ii) & & (1+\| u_n\| _{D^{1,p}})\|J'(u_n)\|_{(D^{1,p})^{*}}\rightarrow 0, \end{array} \label{C1}$$ 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 $01$, the sequence $(u_n)$ is bounded, since $\mu >\frac Np(q-p)$ is equivalent to $tq0$. 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 $\mu 0$ 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)|0$, such that $\int_{\{x:|t\varphi (x)|0,\quad \forall x\in {\Bbb R}^N,\forall u\neq 0.$ 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}% ). \begin{thebibliography}{99} \bibitem{Allegretto-Huang} W. Allegretto and Y. X. Huang, {\em Eigenvalue of the indefinite-weight p-Laplacian in weighted spaces}, Funkcialaj Ekvacioj 38(1995), 233-242. \bibitem{Bartolo-Benci-Fortunato} P. Bartolo, V. Benci and D. Fortunato, {\em Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity}, Nonlinear Anal. T.M.A. 7(1983), 981-1012. \bibitem{Ben-Troestler-Willem} A. Ben-Naoum, C. Troestler and M. 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