\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{equation}
\begin{gathered}
  -\Delta u=f(u),\\
  u\in H_0^1(\Omega).\\
\end{gathered}\label{e1.1}
\end{equation}
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
\begin{equation}
\alpha:=\lim_{|u|\to\infty}\frac{f(u)}{u}\label{e1.2}
\end{equation}
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{equation}
 \begin{gathered}
  -\Delta u-\lambda_k u+g(u)=0,\\
  u\in H_0^1(\Omega).\\
  \end{gathered}\label{e1.3}
\end{equation}
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
\begin{equation}
g'(0)=\sup\{g'(t): t\in\mathbb{R}\}\label{e1.4},
\end{equation}
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 $]c_1,c_2[$  ($-\infty\leq
c_1<c_2\leq+\infty$) if any sequence
$\{u_k\}\subset J^{-1}(]c_1,c_2[)$,
such that $\{J(u_k)\}$ is bounded and $J'(u_k)\to0$, we have
either
\begin{itemize}
\item[(i)]  $\{u_k\}$ is bounded and possesses a
convergent subsequence, or

\item[(ii)] for all $c\in]c_1,c_2[$, there exists
$\sigma, R, \alpha>0$ 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
$$
J(u)\geq c_0, \forall u\in S_\rho\cap H^+;\quad
J(u)<c_\infty, \forall u\in H^-.
$$

\item[(3)] $J$ is even.
\end{itemize}
Then if $\dim H^->1+\operatorname{codim} H^+$,
 $J$ 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
\begin{equation}
J(0)<c_0\leq c_k\leq c_\infty,
\quad k=2,\dots, \dim H^--\operatorname{codim} H^+.\label{e3.1}
\end{equation}
If for every $k$, $c_k\neq c_{k+1}$, we obtain the conclusion of
Theorem \ref{thm3.2}.
Assume now
$c=c_k=\dots=c_{k+r}$ with $r\geq1$ and
$k+r\leq\dim H^--\operatorname{codim} H^+$.
Then as in the proof to \cite[Theorem 2.9]{QL}, we have
\begin{equation}
i(K_c\cap S)\geq r+1\geq2\label{e3.2}
\end{equation}
Now from proposition \ref{prop3.3} and \eqref{e3.1}, we
deduce that
\begin{equation}
0\not\in K_c\cap S.\label{e3.3}
\end{equation}
Since a finite set (not containing 0) has genus 1, we deduce
from \eqref{e3.2} and \eqref{e3.3} that
$K_c$ above contains infinitely many sign-changing critical points.
Therefore, $J$ has at least $m:=\dim H^--\operatorname{codim} H^+-1$
distinct pairs of sign-changing critical points in
$X\backslash P\cup(-P)$
with critical values belong to $[c_0,c_\infty]$.

If $\operatorname{codim} H^+=0$, we consider $c_j$ for $j\geq2$.
As above arguments
$J(0)<c_0\leq c_2\leq c_3\leq\dots\leq c_{\dim H^-}\leq
c_\infty$ and if
$c:=c_j=\dots= c_{j+l}$ for $2\leq j\leq j+l\leq\dim H^-$ with
$l\geq1$, then $i(K_c\cap S)\geq l+1\geq2$.
Therefore, $J$ has at least $\dim H^--1$ pairs of sign-changing
critical points with values belong to
$[c_0,c_\infty]$.
\end{proof}

We remark that Theorem \ref{thm3.1} above can also be proved by the
pseudo-index theory as Theorem \ref{thm3.2}.

\section{Proof of Theorems \ref{thm1.1}--\ref{thm1.3}}

We shall apply the abstract results of section 3 to
problem \eqref{e1.3}.
Let $H:=H_0^1(\Omega), X:=C_0^1(\Omega)$. Clearly the solutions of
 \eqref{e1.3} are the critical points of the functional
\begin{equation}
J(u)=\frac{1}{2}(\|u\|^2-\lambda_k|u|^2)
+\int_\Omega G(u)dx,\label{e4.1}
\end{equation}
where $|\cdot|$ denotes the norm in $L^2(\Omega)$, then
$J\in C^1(H,\mathbb{R})$. We denote by $M_j$ the eigenspace
corresponding to the eigenvalue $\lambda_j$.
If $m\geq0$ is an integer number, set
$$
H^-(m)=\oplus_{j\leq m}M_j,
$$
and $H^+(m)$ the closure in $H_0^1(\Omega)$ of the linear space
spanned by $\{M_j\}_{j\geq m}$.
 Clearly $H^+(m)\cap H^-(m)=M_m$.

\begin{proposition} \label{prop4.1}
If {\rm (G1), (G2)} hold, then the functional $J$
defined by \eqref{e4.1} satisfies the condition (C) in
$]0,+\infty[$.
\end{proposition}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
If $G(0)=0$, then by (G3), $G$ takes its
minimum at 0, so $g(0)=0$ and 0 is a solution of \eqref{e1.3}.
We assume that $G(0)>0$. 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:
\begin{equation}
\lambda_1-\lambda_k+g'(0)>0, \label{e4.1b}
\end{equation}
$\lambda_k\neq\lambda_1$ and there exists
$\lambda_h\in\sigma(-\Delta)$ with
$\lambda_2\leq\lambda_h\leq\lambda_k$ such that
\begin{equation}
\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}
\end{equation}
Under \eqref{e4.1}, there exist three positive constants
$\rho<R, \gamma$ such that
\begin{gather*}
J(u)\geq J(0)+\gamma,\quad u\in S_\rho;
J(e)\leq J(0)+\frac\gamma2,\quad e\in M_1\cap S_\rho.
\end{gather*}
Since $J(0)=G(0)\cdot|\Omega|\geq0$ ($|\Omega|$ is the Lebesgue
measure of $\Omega$), we have
$$
0<J(0)+\frac\gamma2<J(0)+\gamma.
$$
Fix $e\in M_1\cap S_\rho$, set
$$
A=S_\rho,\quad B=\{te: t\in[0,R]\}.
$$
Then by Example \ref{examp3.1}, $A$ and $\partial B$ link and
$J$ is bounded on $B$. Moreover by Proposition \ref{prop4.1},
$J$ satisfies condition (C) in $]0,+\infty[$.
Then by Theorem \ref{thm3.1}, $J$ possesses a critical point
$u_0$ such that $J(u_0)\geq J(0)+\gamma$. So $u_0$ is a
sign-changing solution to  \eqref{e1.3}.

Under \eqref{e4.2} similar arguments to that above, we get
\begin{gather*}
J(u)\geq J(0)+\gamma,\quad u\in H^+(h)\cap S_\rho;\
J(u)\leq J(0)+\frac\gamma2,\quad u\in \partial B(h, R).
\end{gather*}
where $B(h,R)=\{u+te: u\in H^-(h-1)\cap B_R, \; e\in
M_h\cap S_1,\quad 0\leq t\leq R\}$.
Set
$$
A=H^+(h)\cap S_\rho, \quad B=B(h,R).
$$
Then by Example \ref{examp3.2}, $A$ and $\partial B$
link and $J$ is bounded on $B$. Moreover by Proposition \ref{prop4.1},
$J$ satisfies condition (C). Using Theorem \ref{thm3.1}, we can conclude
that $J$ possesses a sign-changing critical point $u_0$
with $J(u_0)\geq J(0)+\gamma$.
\end{proof}


\begin{remark} \label{rmk4.2} \rm
 If $g'(0)=0$; i.e., resonance at 0 is allowed, then
by using an argument similar to that in the proof of
Theorem \ref{thm1.2},
problem \eqref{e1.3} still has at least a sign-changing solution under
these conditions:
Let $g(0)=0$. Assume that (G1), (G2) hold and
$$
G(t)>0,\quad \forall t\neq0,\quad G(0)=0.
$$
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.


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\end{document}