\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 17, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/17\hfil Sing-changing solutions] {Sing-changing solutions for nonlinear problems with strong resonance} \author[A. Qian \hfil EJDE-2012/17\hfilneg] {Aixia Qian} \address{Aixia Qian \newline School of Mathematic Sciences, Qufu Normal University\\ Qufu, Shandong 273165, China} \email{qaixia@amss.ac.cn} \thanks{Submitted June 14, 2011. Published January 26, 2012.} \subjclass[2000]{35J65, 58E05} \keywords{Critical point theory; strong resonance; index theory; \hfill\break\indent Cerami condition} \begin{abstract} Using critical point theory and index theory, we prove the existence and multiplicity of sign-changing solutions for some elliptic problems with strong resonance at infinity, under weaker conditions than in the references. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we consider the equation $$\begin{gathered} -\Delta u=f(u),\\ u\in H_0^1(\Omega).\\ \end{gathered}\label{e1.1}$$ where $\Omega$ is a bounded domain in $\mathbb{\mathbb{R}}^n$ with smooth boundary $\partial\Omega$. We assume that $f$ is asymptotically linear; i.e., $\lim_{|u|\to\infty}\frac{f(u)}{u}$ exists. By setting $$\alpha:=\lim_{|u|\to\infty}\frac{f(u)}{u}\label{e1.2}$$ we can write $f(u)=\alpha u-g(u)$, where $$\frac{g(u)}{u}\to0\quad\text{as } |u|\to\infty.$$ We Denote by $\lambda_1<\lambda_2<\dots<\lambda_j<\dots$ the distinct eigenvalues sequence of $-\Delta$ with the Dirichlet boundary conditions. We say that problem \eqref{e1.1} is resonant at infinity if $\alpha$ in \eqref{e1.2} is an eigenvalue $\lambda_k$. The situation when $$\lim_{|u|\to\infty}g(u)=0\quad\text{and}\quad \lim_{|u|\to\infty}\int_0^u g(t)dt=\beta\in\mathbb{R}$$ is what we call strong resonance. Now we present some results of this paper. We write \eqref{e1.1} in the form $$\begin{gathered} -\Delta u-\lambda_k u+g(u)=0,\\ u\in H_0^1(\Omega).\\ \end{gathered}\label{e1.3}$$ We assume that $g$ is a smooth function satisfying the following conditions. \begin{itemize} \item[(G1)] $g(t) t\to 0$ as $|t|\to\infty$; \item[(G2)] the real function $G(t)=\int_0^t g(s)ds$ is well defined and $G(t)\to 0$ as $t\to+\infty$. \item[(G3)] $G(t)\geq 0$ for all $t\in\mathbb{R}$. \end{itemize} \begin{theorem} \label{thm1.1} If {\rm (G1)--(G3)} hold, then \eqref{e1.1} has at least one solution. \end{theorem} Since $0$ is a particular point, we cannot ensure those solutions are nontrivial without additional conditions. \begin{theorem} \label{thm1.2} Let $g(0)=0$, and suppose that {\rm (G1)--(G3)} hold, and $$g'(0)=\sup\{g'(t): t\in\mathbb{R}\}\label{e1.4},$$ then \eqref{e1.3} has at least one sign-changing solution. \end{theorem} \begin{theorem} \label{thm1.3} Assume {(G1)--(G3)} hold and $g$ is odd and $G(0)\geq 0$. Moreover suppose that there exists an eigenvalue $\lambda_h<\lambda_k$ such that $$g'(0)+\lambda_h-\lambda_k>0.$$ Then \eqref{e1.3} possess at least $m=\dim(M_h\oplus\dots\oplus M_k)-1$ distinct pairs of sign-changing solutions ($M_j$ denotes the eigenspace corresponding to $\lambda_j$). \end{theorem} \begin{remark} \label{rmk1.2} \rm The references show only the existence of solutions to \eqref{e1.3}, while we obtain its sign-changing solutions under weaker conditions. \end{remark} The resonance problem has been widely studied by many authors using various methods; see \cite{BBF,LW,RD,Q,S1,S2} and the references therein. We will use critical point theory and pseudo-index theory to obtain sign-changing solutions for the strong resonant problem \eqref{e1.3}. We also allow the case in which resonance also occurs at zero. In section 2, we give some preliminaries, which are fundamental in our paper. In section 3, we give some abstract critical point theorems, which are used to prove above theorems in this paper. In section 3, by using the above theorems, we prove the existence and multiplicity of sign-changing solutions. \section{Preliminaries} We denote by $X$ a real Banach space. $B_R$ denotes the closed ball in $X$ centered at the origin and with radius $R>0$. $J$ is a continuously Fr\`echet differentiable map from $X$ to $\mathbb{R}$; i.e., $J\in C^1(X,\mathbb{R})$. In the literature, deformation theorems have been proved under the assumption that $J\in C^1(X,\mathbb{R})$ satisfies the well known Palais-Smale condition. In problems which do not have resonance at infinity, the (PS) condition is easy to verify. On the other hand, a weaker condition than the (PS) condition is needed to study problems with strong resonance at infinity. \begin{definition} \label{def2.1} \rm We say that $J\in C^1(X,\mathbb{R})$ satisfies the condition (C) in $]c_1,c_2[$ ($-\infty\leq c_10$ such that $[c-\sigma,c+\sigma]\subset]c_1,c_2[$ and for all $u\in J^{-1}([c-\sigma,c+\sigma])$, $\|u\|\geq R:\|J'(u)\|\|u\|\geq\alpha$. \end{itemize} \end{definition} In \cite{BBF,M,RD}, deformation theorems are obtained under the condition (C). For $c\in\mathbb{R}$, denote $$A_c=\{u\in X: J(u)\leq c\},\quad K_c=\{u\in X: J'(u)=0, J(u)=c\}.$$ \begin{proposition} \label{prop2.2} Let $X$ be a real Banach space, and let $J\in C^1(X,\mathbb{R})$ satisfy the condition (C) in $]c_1,c_2[$. If $c\in]c_1,c_2[$ and $N$ is any neighborhood of $K_c$, there exists a bounded homeomorphism $\eta$ of $X$ onto $X$ and constants $\bar{\varepsilon}>\varepsilon>0$, such that $[c-\bar{\varepsilon},c+\bar{\varepsilon}]\subset]c_1,c_2[$ satisfying the following properties: \begin{itemize} \item[(i)] $\eta(A_{c+\varepsilon}\backslash N)\subset A_{c-\varepsilon}$. \item[(ii)] $\eta(A_{c+\varepsilon})\subset A_{c-\varepsilon}$, if $K_c=\emptyset$. \item[(iii)] $\eta(x)=x$, if $x\not\in J^{-1}([c-\bar{\varepsilon},c+\bar{\varepsilon}])$. \end{itemize} Moreover, if $G$ is a compact group of (linear) unitary transformation on a real Hilbert space $H$, then \begin{itemize} \item[(vi)] $\eta$ can be chosen to be $G$-equivariant, if the functional $J$ is $G$-invariant. Particularly, $\eta$ is odd if the functional $J$ is even. \end{itemize} \end{proposition} \section{Abstract critical point theorems} In this article, we shall obtain solutions to \eqref{e1.3} by using the linking type theorem. Its different definitions can be seen in \cite{ST,Z} and references therein. \begin{definition} \label{def3.1}\rm Let $H$ be a real Hilbert space and $A$ a closed set in $H$. Let $B$ be an Hilbert manifold with boundary $\partial B$, we say $A$ and $\partial B$ link if \begin{itemize} \item[(i)] $A\cap\partial B=\emptyset$; \item[(ii)] If $\phi$ is a continuous map of $H$ into itself such that $\phi(u)=u$ for all $u\in\partial B$, then $\phi(B)\cap A\neq\emptyset$. \end{itemize} Typical examples can be found in \cite{BBF,QL,R,Z}. \end{definition} \begin{example} \label{examp3.1}\rm Let $H_1, H_2$ be two closed subspaces of $H$ such that $$H=H_1\oplus H_2,\quad \dim H_2<\infty.$$ Then if $A=H_1$, $B=B_R\cap H_2$, then $A$ and $\partial B$ link. \end{example} \begin{example} \label{examp3.2} \rm Let $H_1, H_2$ be two closed subspaces of $H$ such that $H=H_1\oplus H_2$, $\dim H_2<\infty$, and consider $e\in H_1$, $\|e\|=1$, $0<\rho< R_1, R_2$, set $$A=H_1\cap S_\rho,\quad B=\{u=v+te: v\in H_2\cap B_{R_2}, 0\leq t\leq R_1\}.$$ Then $A$ and $\partial B$ link. \end{example} Let $X\subset H$ be a Banach space densely embedded in $H$. Assume that $H$ has a closed convex cone $P_H$ and that $P:=P_H\cap X$ has interior points in $X$. Let $J\in C^1(H,\mathbb{R})$. In \cite{QL}, they construct the pseudo-gradient flow $\sigma$ for $J$, and have the following definition. \begin{definition} \label{def3.2} Let $W\subset X$ be an invariant set under $\sigma$. $W$ is said to be an admissible invariant set for $J$ if \begin{itemize} \item[(a)] $W$ is the closure of an open set in $X$; \item[(b)] if $u_n=\sigma(t_n,v)\to u$ in $H$ as $t_n\to\infty$ for some $v\not\in W$ and $u\in K$, then $u_n\to u$ in $X$; \item[(c)] If $u_n\in K\cap W$ is such that $u_n\to u$ in $H$, then $u_n\to u$ in $X$; (d) For any $u\in\partial W\backslash K$, we have $\sigma(t,u)\in \mathring{W}$ for $t>0$. \end{itemize} \end{definition} Now let $S=X\backslash W$, $W=P\cup(-P)$. As the similar proof to that in \cite{QL}, the $W$ is an admissible invariant set for $J$ in the following section 4. We define $\phi^*=\{\Gamma(t,x):[0,1]\times X\to X \text{ continuous in the X-topology and }\Gamma(t,W)\subset W\}.$ In \cite{Z}, a new linking theorem is given under the condition (PS). Since the deformation is still hold under the condition (C), the following theorem holds. \begin{theorem} \label{thm3.1} Suppose that $W$ is an admissible invariant set of $J$ and that $J$ is in $C^1(H,\mathbb{R})$ such that \begin{itemize} \item[(1)] $J$ satisfies condition (C) in $]0,+\infty[$; \item[(2)] There exist a closed subset $A\subset H$ and a Hilbert manifold $B\subset H$ with boundary $\partial B$ satisfying \begin{itemize} \item[(a)] there exist two constants $\beta>\alpha\geq0$ such that $$J(u)\leq\alpha,\; \forall u\in\partial B;\quad J(u)\geq\beta,\; \forall u\in A;$$ i.e., $a_0:=\sup_{\partial B}J\leq b_0:=\inf_{A}J$. \item[(b)] $A$ and $\partial B$ link; \item[(c)] $\sup_{u\in B}J(u)<+\infty$. \end{itemize} \end{itemize} Then a critical value of $J$ is given by $$a^*=\inf_{\Gamma\in\phi^*}\sup_{\Gamma([0,1],A)\cap S}J(u).$$ Furthermore, assuming that $0\not\in K_{a^*}$, we have $K_{a^*}\cap S\neq\emptyset$ if $a^*>b_0$, and $K_{a^*}\cap A\neq\emptyset$ if $a^*=b_0$. \end{theorem} In this article, we shall consider the symmetry given by a $\mathbb{Z}_2$ action, more precisely even functionals. \begin{theorem} \label{thm3.2} Suppose $J\in C^1(H,\mathbb{R})$ and the positive cone $P$ is an admissible invariant for $J$, $K_c\cap\partial P=\emptyset$ for $c>0$, such that \begin{itemize} \item[(1)] $J$ satisfies condition (C) in $]0,+\infty[$, and $J(0)\geq0$; \item[(2)] There exist two closed subspace $H^+, H^-$ of $H$, with co$\dim H^+<+\infty$ and two constants $c_\infty>c_0>J(0)$ satisfying $$J(u)\geq c_0, \forall u\in S_\rho\cap H^+;\quad J(u)1+\operatorname{codim} H^+, J possesses at least m:=\dim H^- -\operatorname{codim} H^+-1 (m:=\dim H^--1 resp.) distinct pairs of critical points in X\backslash P\cup(-P) with critical values belong to [c_0,c_\infty]. \end{theorem} The above theorem locates the critical points more precisely than \cite[Theorem 3.3]{BBF,QL} and the references therein. We shall use pseudo-index theory to prove Theorem \ref{thm3.2}. First, we need the notation of genus and its properties, see \cite{QL,R}. Let $\Sigma_X=\{A\subset X: A\text{ is closed in }X, A=-A\}.$ We denote by i_X(A) the genus of A in X. \begin{proposition} \label{prop3.2} Assume that A, B\in \Sigma_X, h\in C(X,X) is an odd homeomorphism, then \begin{itemize} \item[(i)] i_X(A)=0 if and only if A=\emptyset; \item[(ii)] A\subset B\Rightarrow i_X(A)\leq i_X(B) (monotonicity); \item[(iii)] i_X(A\cup B)\leq i_X(A)+i_X(B) (subadditivity); \item[(iv)] i_X(A)\leq i_X(\overline{h(A)}) (supervariancy); \item[(v)] if A is a compact set, then i_X(A)<+\infty and there exists \delta>0 such that i_X(N_\delta(A))=i_X(A), where N_\delta(A) denotes the closed \delta-neighborhood of A (continuity); \item[(vi)] if i_X(A)>k, V is a k-dimensional subspace of X, then A\cap V^\bot\neq\emptyset; \item[(vii)] if W is a finite dimensional subspace of X, then i_X(h(S_\rho)\cap W)=\dim W. \item[(viii)] Let V, W be two closed subspace of X with co\dim V<+\infty, \dim W<+\infty. Then if h is bounded odd homeomorphism on X, we have $i_X(W\cap h(S_\rho\cap V))\geq\dim W-\operatorname{codim} V.$ \end{itemize} \end{proposition} The above proposition is still true when we replace \Sigma_X by \Sigma_H with obvious modifications. \begin{proposition}[\cite{QL}] \label{prop3.3} If A\in \Sigma_X with 2\leq i_X(A)<\infty, then A\cap S\neq\emptyset. \end{proposition} \begin{proposition} \label{prop3.4} Let A\in \Sigma_H, then A\cap X\in\Sigma_X and i_H(A)\geq i_X(A\cap X). \end{proposition} \begin{definition}[\cite{BBF,QL}] \label{def3.3} \rm Let I=(\Sigma, \mathcal{H}, i) be an index theory on H related to a group G, and B\in\Sigma. We call a pseudo-index theory (related to B and I) a triplet$$ I^*=(B,\mathcal{H}^*,i^*) $$where \mathcal{H}^*\subset\mathcal{H} is a group of homeomorphism on H, and i^*:\Sigma\to\mathbb{N}\cup\{+\infty\} is the map defined by$$ i^*(A)=\min_{h\in\mathcal{H}^*}i(h(A)\cap B). \end{definition} \begin{proof}[Proof of Theorem \ref{thm3.2}] Consider the genus I=(\Sigma,\mathcal{H}, i) and the pseudo-index theory relate to I and B=S_\rho\cap H^+, I^*=(S_\rho\cap H^+,\mathcal{H}^*, i^*), where \begin{align*} \mathcal{H}^*=\{& h\text{ is an odd bounded homeomorphism on H and}\\ & h(u)=u \text{ if }u\not\in J^{-1}(]0,+\infty[)\}. \end{align*} Obviously conditions \cite[(a_1), (a_2) of theorem 2.9]{BBF} are satisfied with a=0, b=+\infty and b=S_\rho\cap H^+. Now we prove that condition (a_3) is satisfied with \bar{A}=H^-. It is obvious that \bar{A}\subset J^{-1}(]-\infty,c_\infty]), and by property (iv) of genus, we have \begin{align*} i^*(\bar{A})=i^*(H^-) &= \min_{h\in\mathcal{H}^*}i(h(H^-)\cap S_\rho\cap H^+)\\ &= \min_{h\in\mathcal{H}^*}i(H^-\cap h^{-1}(S_\rho\cap H^+)). \end{align*} Now by (viii) of Proposition \ref{prop3.2}, we have $i(H^-\cap h^{-1}(S_\rho\cap H^+))\geq\dim H^--\operatorname{codim} H^+.$ Therefore, we have $i^*(\bar{A})\geq\dim H^--\operatorname{codim}H^+.$ Then by \cite[Theorem 2.9]{QL} and Proposition \ref{prop3.3} above, the numbers $c_k=\inf_{A\in\Sigma_k}\sup_{u\in A\cap S}J(u), \quad k=2,\dots,\dim H^--\operatorname{codim} H^+.$ are critical values of J and J(0)0. Similar proof to that in \cite{BBF}, there exist R,\gamma>0 such that \begin{gather*} J(u)\geq\gamma, \quad u\in H^+(k+1);\\ J(u)\leq\frac \gamma2,\quad u\in H^-(k)\cap S_R. \end{gather*} Let \partial B=H^-(k)\cap S_R, A=H^+(k+1), then by Example \ref{examp3.1} we have that \partial B and A link, and J is bounded on B=H^-(k)\cap B_R. Moreover by Proposition \ref{prop4.1}, J satisfies condition (C) in ]0,+\infty[. So the conclusion of Theorem \ref{thm1.1} follows by Theorem \ref{thm3.1}. \end{proof} Note that if J(0)=0, then the solutions obtained in Theorem \ref{thm1.1} are sign-changing solutions. \begin{proof}[Proof of Theorem \ref{thm1.2}] Since g(0)=0, u(x)=0 is a solution of \eqref{e1.3}. In this case, we are interested in finding the existence of sign-changing solutions to \eqref{e1.3}. The case g(t)=0 for all t\in\mathbb{R} is trivial. We assume that g(t)\neq 0 for some t. Then it is easy to see that (G2), (G3) and \eqref{e1.4} imply g'(0)>0. Similar proof to that in \cite[Theorem 5.1]{BBF}, each of the following holds: $$\lambda_1-\lambda_k+g'(0)>0, \label{e4.1b}$$ \lambda_k\neq\lambda_1 and there exists \lambda_h\in\sigma(-\Delta) with \lambda_2\leq\lambda_h\leq\lambda_k such that $$\lambda_h-\lambda_k+g'(0)>0,\quad \frac12(\lambda_{h-1}-\lambda_k)t^2+G(t)\leq G(0)\quad \forall t\in\mathbb{R}.\label{e4.2}$$ Under \eqref{e4.1}, there exist three positive constants \rho0,\quad \forall t\neq0,\quad G(0)=0. Moreover suppose that either of the following holds \begin{gather*} \lambda_k=\lambda_1;\\ \lambda_k\neq\lambda_1\text{ and } \frac12(\lambda_{k-1}-\lambda_k)t^2+G(t)\leq0\quad \text{for all }t\in\mathbb{R}. \end{gather*} \end{remark} \begin{proof}[Proof of Theorem \ref{thm1.3}] By \cite[Proposition 3.1 and Lemma 5.3]{BBF}, the assumptions of Theorem \ref{thm3.2} are satisfied with $$H^+=H^+(h),\quad H^-=H^-(k).$$ Thus there exist at least $\dim H^--\operatorname{codim} H^+-1=\dim\{M_h\oplus\dots M_k\}-1$ distinct pairs of sign-changing solutions of \eqref{e1.3}. \end{proof} \begin{remark} \label{rmk4.3} \rm We also allow resonance at zero in problem \eqref{e1.3}. By using \cite[Theorem 3.2 and Lemma 5.4]{BBF}, we have: Assume that $g$ is odd and (G1) (G2) are satisfied. Suppose moreover $G(t)>0$ for all $t\neq 0$ and $G(0)=0$. Then \eqref{e1.3} possesses at least $\dim M_k-1$ distinct pairs of sign-changing solutions. ($M_k$ denotes the eigenspace corresponding to $\lambda_k$ with $k\geq2$) \end{remark} \subsection*{Acknowledgements} The author is grateful to the anonymous referee for his or her suggestions. This research was supported by grants 11001151 and 10726003 from the Chinese National Science Foundation, Q2008A03 from the National Science Foundation of Shandong, 201000481301 from the Science Foundation of China Postdoctoral, and from the Shandong Postdoctoral program. \begin{thebibliography}{99} \bibitem{BBF} P. Bartolo, V. Benci, D. Fortunato; \emph{Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity}, Nonl. Anal. 7(1983), 981-1012. \bibitem{LW} Z. L. Liu, Z. Q. Wang; \emph{Sign-changing solutions of nonlinear elliptic equations}, Front. Math. China. 3(2008), 221-238. \bibitem{M} A. M. Mao; \emph{Periodic solutions for a class of non-autonomous Hamiltonian systems}. Nonl. Anal. 61(2005), 1413-1426. \bibitem{RD} R. Molle, D. 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