\documentclass[twoside]{article} \usepackage{amssymb,amsmath} \pagestyle{myheadings} \markboth{\hfil Existence of solutions \hfil EJDE--2001/66} {EJDE--2001/66\hfil Shibo Liu \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 66, pp. 1--6. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Existence of solutions to a superlinear $p$-Laplacian equation % \thanks{ {\em Mathematics Subject Classifications:} 49J35, 35J65, 35B34. \hfil\break\indent {\em Key words:} $p$-Laplacian, critical group. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted August 21, 2001. Published October 11, 2001.} } \date{} % \author{Shibo Liu} \maketitle \begin{abstract} Using Morse theory, we establish the existence of solutions to the equation $-\Delta_p u = f(x,u)$ with Dirichlet boundary conditions. We assume that $\int_0^s f(x,t)\,dt$ lies between the first two eigenvalues of the $p$-Laplacian. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} Consider the Dirichlet problem for the $p$-Laplacian $(p>1)$, $$\begin{gathered} -\Delta _pu=f(x,u), \quad \text{in }\Omega , \\ u=0, \quad \text{on }\partial \Omega . \end{gathered} \label{P}$$ Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $% \partial \Omega$, and $-\Delta _pu$ is the $p$-Laplacian: $-\Delta _pu:=\mathop{\rm div}(|\nabla u| ^{p-2}\nabla u)$. We assume that $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a Carath\'eodory function with subcritical growth; that is, \begin{enumerate} \item[F1)] The inequality $|f(x,u)|\leq C( 1+|u|^{q-1})$ holds for all $u\in \mathbb{R}$, $x\in \Omega$, and for some positive constant $C$, where $1\leq q<\frac{Np}{N-p}$ if $N\geq p+1$, and $1\leq q<\infty$ if $1\leq N0$ is a simple eigenvalue, and that $\sigma (-\Delta _p)\cap ( \lambda _1,\lambda _2) =\emptyset$, where $\sigma (-\Delta _p)$ is the spectrum of $-\Delta _p$, (cf. \cite{T}). We shall assume the following conditions: \begin{enumerate} \item[F2)] There exist $r>0$, $\bar \lambda \in ( \lambda_1,\lambda _2)$ such that $|u| \leq r$ implies $\lambda _1| u| ^p\leq pF( x,u) \leq \bar \lambda | u| ^p,$ \item[F3)] There exist $\theta >p$, $M>0$ such that $| u| \geq M$ implies $0<\theta F( x,u) \leq u f( x,u) .$ \end{enumerate} Now, we are ready to state our main result. \begin{theorem} \label{thm}Assume (F1), (F2), and (F3). Then (\ref{P}) has a nontrivial weak solution in $W_0^{1,p}(\Omega )$. \end{theorem} There are many papers devoted to the existence of solutions of (\ref{P}); see for example \cite{AM, CM, FanL}. In these papers, the main tool is the minmax argument. However, it seems difficult to use the minmax argument in our situation. Thus we will use a different approach: Morse theory \cite{Cha}. To the best of our knowledge, \cite{LiuS} is the only work using Morse theory to obtain the solvability of $p$-Laplacian equations. Our work is motivated by \cite{LiuS}. \section{Proof of main theorem} In this section we give the proof of Theorem \ref{thm}. Let $E$ denote the Sobolev space $W_0^{1,p}( \Omega )$, and $\Vert .\Vert$ denote the norm in $E$. For $\Phi$ a continuously Fr\'echet differentiable map from $E$ to $\mathbb{R}$, let $\Phi '(u)$ denote its Fr\'echet derivative. As stated in Section 1, weak solutions $u\in W_0^{1,p}(\Omega )$ of (\ref {P}) are the critical points of the $C^1$ functional $\Phi (u)=\frac 1p\int |\nabla u|^p\,dx-\int F(x,u)\,dx\,.$ We will try to find a nontrivial critical point of the functional $\Phi$. First we state the following lemmas. \begin{lemma} \label{lem1} Under conditions (F1) and (F3), the functional $\Phi$ satisfies the Palais-Smale condition. \end{lemma} \paragraph{Proof} Assume $( u_n) \subset E$, $| \Phi( u_n) | \leq B$ for some $B\in \mathbb{R}$, and $\Phi'( u_n) \rightarrow 0$. Let $d:=\sup_n\Phi ( u_n)$. Then by (F3) we have \begin{eqnarray*} \theta d+\| u_n\| &\geq &\theta \Phi ( u_n) +\left\langle \Phi '( u_n) ,u_n\right\rangle \\ &=&( \frac \theta p-1) \| u_n\| ^p-\int_{| u_n| \geq M}[ \theta F( x,u_n) -f( x,u_n)u_n] \\ && -\int_{| u_n| \leq M}[ \theta F( x,u_n) -f( x,u_n) u_n] \\ &\geq &( \frac \theta p-1) \| u_n\| ^p-\int_{| u_n| \leq M}[ \theta F( x,u_n) -f( x,u_n) u_n] \\ &\geq &( \frac \theta p-1) \| u_n\| ^p-D,\quad \text{for some }D\in \mathbb{R}. \end{eqnarray*} Thus $( u_n)$ is bounded in $E$. Up to a subsequence, we may assume that $u_n\rightharpoonup u$ in $E$. Now because of condition (F1), a standard argument shows that $u_n\rightarrow u$ in $E$ and the proof is complete. \hfill$\diamondsuit$\smallskip Let $V=\mathop{\rm span}{ \phi _1}$ be the one-dimensional eigenspace associated to $\lambda _1$, where $\phi _1>0$ in $\Omega$ and $\| \phi _1\| =1$. Taking a subspace $W\subset E$ complementing $V$, that is $E=V\oplus W$. Obviously the genus of $W\backslash { 0}$ satisfies $\gamma ( W\backslash { 0} ) \geq 2$. Therefore, by the variational characterization of $\lambda _2$, for $\forall u\in W$, $\int | \nabla u| ^p\geq \lambda _2\int | u| ^p.$ \begin{lemma} \label{lem2} Under Assumption (F2), the functional $\Phi$ has a local linking at the origin with respect to $E=V\oplus W$. That is, there exists $\rho >0$, such that \begin{gather*} \Phi ( u) \leq 0,\quad u\in V, \; \| u\| \leq \rho , \\ \Phi ( u) >0, \quad u\in W,\; 0<\| u\| \leq \rho . \end{gather*} \end{lemma} The proof of this lemma can be found in \cite[Lemma 3.3]{LiuS}. \smallskip For a $C^1$-functional $\Phi :E\rightarrow \mathbb{R}$ and $u$ an isolate critical point of $\Phi$, $\Phi (u) =c$, we define the critical group of $\Phi$ at $u$ as $C_q( \Phi ,u) :=H_q( \Phi _c,\Phi _c\backslash \{u\} ) .$ Where $H_q( X,Y)$ is the $q$-th homology group of the topological pair $( X,Y)$ over the ring $\mathbb{Z}$. Since $\dim V=1<+\infty$, from Lemma \ref{lem2} and Theorem 2.1 in \cite {Liu}, we have \begin{lemma} \label{lem3} Under assumption (F2), $0$ is a critical point of $\Phi$ and $C_1( \Phi ,0) \neq 0$. \end{lemma} To find a nontrivial critical point of $\Phi$, we investigate the behavior of $\Phi$ near infinity. \begin{lemma} \label{lem4} Under Assumption (F3), there exists a constant $A>0$ such that $\Phi _a\simeq S^\infty, \quad \text{for }a<-A,$ where $S^\infty$ is the unit sphere in $E$. \end{lemma} \paragraph{Proof} Integrating on the inequality of (F2), we obtain a constant $C_1>0$ such that $F( x,t) \geq C_1| t| ^\theta ,\quad\text{ }\quad\text{for } | t| \geq M.$ Thus, for $u\in S^\infty$, we have $\Phi ( tu) \rightarrow -\infty$, as $t\rightarrow+\infty$. Set $A:=\left( 1+\frac 1p\right) M| \Omega | \max_{\bar \Omega \times [ -M,M] }| f( x,u) | +1\,.$ Using (F3) we obtain \begin{eqnarray*} &&\int F\left( x,v\right) -\frac 1p\int vf\left( x,v\right) \\ &=&\int_{\left| v\right| \geq M}F\left( x,v\right) +\int_{\left| v\right| \leq M}F\left( x,v\right) -\frac 1p\int_{\left| v\right| \geq M}vf\left( x,v\right) -\frac 1p\int_{\left| v\right| \leq M}vf\left( x,v\right) \\ &\leq &\left( \frac 1\theta -\frac 1p\right) \int_{\left| v\right| \geq M}vf\left( x,v\right) +\int_{\left| v\right| \leq M}F\left( x,v\right) -\frac 1p\int_{\left| v\right| \leq M}vf\left( x,v\right) \\ &\leq &\left( \frac 1\theta -\frac 1p\right) \int_{\left| v\right| \geq M}vf\left( x,v\right) +\left( 1+\frac 1p\right) M\left| \Omega \right| \max_{\bar \Omega \times \left[ -M,M\right] }\left| f\left( x,u\right) \right| \\ &\leq &\left( \frac 1\theta -\frac 1p\right) \int_{\left| v\right| \geq M}vf\left( x,v\right) +A-1\text{.} \end{eqnarray*} For $a<-A$ and $\Phi ( tu) =\frac{| t| ^p}p-\int F( x,tu) \leq a,\quad (u\in S^\infty ),$ we have \begin{eqnarray*} \frac d{dt}\Phi ( tu) &=&\left\langle \Phi '( tu) ,u\right\rangle =| t| ^{p-2}t-\int uf( x,tu) \\ &\leq &\frac pt\Big\{ \int F( x,tu) -\frac 1p\int tuf( x,tu) +a\Big\} \\ &\leq &\frac pt\Big\{ ( \frac 1\theta -\frac 1p) \int_{| tu| \geq M}tuf( x,tu) +A-1+a\Big\} \\ &\leq &\frac pt\Big\{ ( \frac 1\theta -\frac 1p) \int_{| tu| \geq M}tuf( x,tu) -1\Big\} \\ &\leq &\frac pt\Big\{ ( \frac 1\theta -\frac 1p) C_1\theta \int_{| tu| \geq M}| tu| ^\theta -1\Big\} <0. \end{eqnarray*} % By the Implicit Function Theorem, there is a unique $T\in C( S^\infty ,\mathbb{R})$ such that $\Phi ( T( u) u) =a,\quad\forall u\in S^\infty .$ For $u\neq 0$, set $\tilde T( u) =\frac 1{\| u\| }T( \frac u{\| u\| })$. Then $\tilde T\in C( E\backslash { 0} ,\mathbb{R})$ and for all $u\in E\backslash { 0}$, $\Phi ( \tilde T(u) u) =a$. Moreover, if $\Phi ( u) =a$, then $\tilde T( u) =1$. We define a function $\hat T:E\backslash { 0} \rightarrow \mathbb{R}$ as $\hat T( u) :=\begin{cases} \tilde T( u) , & \text{if }\Phi ( u) \geq a, \\ 1, & \text{if }\Phi ( u) \leq a. \end{cases}$ Since $\Phi ( u) =a$ implies $\tilde T( u) =1$, we conclude that $\hat T\in C( E\backslash \{ 0\} ,\mathbb{R})$. Finally we set $\eta :[ 0,1] \times ( E\backslash {0} ) \rightarrow E\backslash { 0}$ as $\eta ( s,u) =( 1-s) u+s\hat T( u) u.$ It is easy to see that $\eta$ is a strong deformation retract from $E\backslash { 0}$ to $\Phi _a$. Thus $\Phi _a\simeq E\backslash { 0} \simeq S^\infty$ and present proof is complete. \hfill$\diamondsuit$\smallskip We also use the following topological result,which was proved by Perera \cite{Per}. \begin{lemma} \label{lem5} Let $Y\subset B\subset A\subset X$ be topological spaces and $q\in \mathbb{Z}$. If $$H_q( A,B) \neq 0 \quad\text{and}\quad H_q( X,Y) =0$$ then $$H_{q+1}( X,A) \neq 0 \quad\text{or}\quad H_{q-1}( B,Y) \neq 0\,.$$ \end{lemma} Now we can prove the main theorem. \paragraph{Proof of Theorem \ref{thm}} By Lemma \ref{lem1}, $\Phi$ satisfies the Palais-Smale condition. Note that $\Phi ( 0) =0$, from \cite{Cha} Chapter I, Theorem 4.2, there is a $\varepsilon >0$, such that $H_1( \Phi _\varepsilon ,\Phi _{-\varepsilon }) =C_1( \Phi ,0) \neq 0.$ By Lemma \ref{lem4}, for $a<-A$ ($A$ is as in the lemma) we have $\Phi _a\simeq S^\infty$. Since $\dim E=+\infty$, $H_1( E,\Phi _a) =H_1( E,S^\infty ) =0.$ So that Lemma \ref{lem5} yields $H_2( E,\Phi _\varepsilon ) \neq 0\text{\quad or\quad }H_0( \Phi _{-\varepsilon },\Phi _a) \neq 0. \label{cri}$ It follows that $\Phi$ has a critical point $u$ for which $\Phi ( u) >\varepsilon \quad \text{or\quad }-\varepsilon >\Phi ( u) >a\,.$ Therefore, $u$ is a nonzero critical point of $\Phi$, and (\ref{P}) has a nontrivial solution. \paragraph{Remark} Result similar to Lemma \ref{lem4} has been proved (for $p=2$) in \cite{Wa} and \cite{Cha}, under the additional conditions $$f\in C^1( \Omega \times \mathbb{R},\mathbb{R}), \quad f( x,0) =\frac{\partial f(x,t) }{\partial t}\Big| _{t=0}=0\,.$$ From these two references, we have obtained the motivation for this paper. \begin{thebibliography}{0} \bibitem{AM} A. R. El Amrouss \& M. Moussaoui, \newblock Minimax principles for critical-point theory in applications to quasilinear boundary-value problems, \newblock \textit{Electron. J. Diff. Eqns.,, 2000(2000),} No. 18, 1--9. \bibitem{T} A. Anane \& N. Tsouli, \newblock On the second eigenvalue of the $p$-Laplacian, \newblock \textit{Nonlinear Partial Differential Equations, Pitman Research Notes\/} 343(1996), 1--9. \bibitem{Cha} K. C. Chang, \newblock Infinite dimensional Morse theory and multiple solution problems, Birkh\"auser, Boston, 1993. \bibitem{CM} D. G. Costa \& C. A. Magalh$\tilde a$es, \newblock Existence results for perturbations of the p-Laplacian, \newblock \textit{Nonlinear Analysis,\/} 24(1995), 409--418. \bibitem{FanL} X. L. Fan \& Z. C. Li, \newblock Linking and existence results for perturbations of the $p$-Laplacian, \newblock \textit{Nonlinear Analysis\/}, 42(2000), 1413-1420. \bibitem{Liu} J. Q. Liu, \newblock The Morse index of a saddle point, % \newblock \textit{Syst. Sc. \& Math. Sc., 2(1989), 32-39.} \bibitem{LiuS} J. Q. Liu \& J. B. Su, \newblock Remarks on multiple nontrivial solutions for quasi-linear resonant problems, \newblock \textit{J. Math. Anal. Appl., 258(2001), 209-222.} \bibitem{Per} K. Perera, \newblock Critical groups of critical points produced by local linking with applications, \newblock \textit{Abstract and Applied Analysis, 3(1998), 437-446.} \bibitem{Wa} Z. Q. Wang, \newblock On a superlinear elliptic equation, % \newblock \textit{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 8(1991), 43-57.} \end{thebibliography} \medskip \noindent \textsc{Shibo Liu } \newline Institute of Mathematics, \newline Academy of Mathematics and Systems Sciences, \newline Academia Sinica, \newline Beijing, 100080, P. R. China \newline e-mail address: liusb@math08.math.ac.cn \end{document}