\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 134, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/134\hfil Existence of infinitely many solutions] {Existence of infinitely many solutions for elliptic boundary-value problems with nonsymmetrical critical nonlinearity} \author[Geng Di\hfil EJDE-2004/134\hfilneg] {Geng Di} \address{Geng Di \hfill\break Department of Mathematics, South China Normal University\\ Guangzhou, 510631, China} \email{gengdi@scnu.edu.cn} \date{} \thanks{Submitted June 29, 2004. Published November 23, 2004.} \thanks{Supported by the National Natural Foundation of China (No. 10371045) and \hfill\break\indent Guangdong Provincial Natural Science Foundation of China (No. 000671)} \subjclass[2000]{35B50, 35J40} \keywords{Dirichlet problem; critical growth; non-symmetric perturbation; \hfill\break\indent infinitely many solutions} \begin{abstract} In this paper, we study a semilinear elliptic boundary-value problem involving nonsymmetrical term with critical growth on a bounded smooth domain in $\mathbb{R}^n$. We show the existence of infinitely many weak solutions under the presence of some symmetric sublinear term, the corresponding critical values of the variational functional are negative and go to zero. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{prop}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction and Main Results} In the present paper, we consider the following Dirichlet problem, for the Laplace equation, \begin{equation} \label{1.1} \begin{gathered} -\Delta u=g(x,u)+f(x,u),\quad \mbox{in }\Omega\\ u=0,\quad \mbox{ on }\partial\Omega\,, \end{gathered} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$ and $n\geq3$. We assume that the nonlinear term $g(x,u)\in C(\Omega\times\mathbb{R})$ is odd symmetric: \begin{equation} \label{1.2} g(x,-u)=-g(x,u),\quad\mbox{for all }(x,u)\in\Omega\times(-\infty,+\infty). \end{equation} The other nonlinear term $f(x,u)$ in \eqref{1.1} is a non-symmetric perturbation. When $f(x,u)=0$, multiple solutions (usually infinitely many solutions) may be expected. As pointed out by many authors, the symmetry is not necessary to guarantee the multiplicity of solutions for \eqref{1.1}; we refer to Rabinowitz \cite{R}, Struwe \cite{Struwe} and Dong \& Li \cite{DL} and the references therein. Some relevant results can be found in \cite{AAP}, \cite {B} and \cite{C}. In these papers authors assumed that the nonlinear terms did not have critical growth, which enables the Palais-Smale condition to be verified in a simple method on a large scale. There arises a natural question whether there still exist multiple solutions even when the symmetric term is non-critical and the perturbed nonlinearity is critical. In this paper, we will partly answer this question. As a special case of \eqref{1.1}, we consider the problem \begin{equation} \label{1.4} \begin{gathered} -\Delta u=|u|^{p-1}u\pm|u|^{n+2\over n-2},\quad \mbox{in }\Omega\\ u=0,\quad \mbox{on }\partial\Omega. \end{gathered} \end{equation} Then we have the following result. \begin{theorem} \label{Thm1} Suppose $1>p>\max\{0,(n(n-2)-4)/(n(n-2)+4)\}$. Then \eqref{1.4} possesses infinitely many (weak) solutions. \end{theorem} In order to get a more general conclusion, we impose the following assumptions on $g(x,u)$ and $f(x,u)$. \begin{itemize} \item[(G1)] There exist positive constants $C_0$, $C_1$ and a nonnegative constant $C_2$ such that $$ C_0|u|^p\leq|g(x,u)|\leq C_1|u|^p+C_2|u|^q, $$ where $p\in(0,1)$ and $q\in[p,1)$. \item[(G2)] There exists a positive constant $\mu\in\big({1\over2},{1\over1+p}\big)$ such that $$ 0\leq\mu ug(x,u)\leq G(x,u)=\int^u_0g(x,\tau)d\tau,\quad\mbox{for all }(x,u)\in\Omega\times(-\infty,+\infty); $$ \end{itemize} Moreover, we suppose that $f\in C^1(\Omega\times\mathbb{R})$ and satisfies \begin{itemize} \item[(F)] There exists a positive constant $C_3$ such that for all $(x,u)\in\Omega\times(-\infty,+\infty)$, $$ |f(x,u)|\leq C_3|u|^{(n+2)/(n-2)},\quad|f'_u(x,u)|\leq C_3(1+|u|^{4/(n-2)}). $$ \end{itemize} \noindent\textbf{Remark.} The growth of $f(x,u)$ is allowed to be critical, that is, when $1\leq s<2^*=2n/(n-2)$, the embedding of the Sobolev space $H_0^1(\Omega)\hookrightarrow L^s(\Omega)$ is compact; if $s=2^*$, the embedding is only continuous but not compact. The best Sobolev embedding constant as $s=2^*$ is denoted by $S$, namely, \begin{equation} \label{1.5} S=\inf\{\|\nabla u\|^2_2;u\in H^1_0(\Omega)\quad\mbox{and}\quad \|u\|_{2^*}=1\}. \end{equation} We introduce a variational functional for \eqref{1.1} as $$ I(u)={1\over2}\int_\Omega|\nabla u|dx-\int_\Omega G(x,u)dx-\int_\Omega F(x,u)dx\,, $$ where $\|u\|=\sqrt{\int_\Omega|\nabla u|^2dx}$ is the norm in $H_0^1(\Omega)$, and $F(x,u)=\int_0^uf(x,\tau)d\tau$. The weak solutions of \eqref{1.1} are the critical points of the functional $I(u)$. Our main result in this paper is the following theorem. \begin{theorem} \label{Thm2} Suppose that the exponent $p$ in the assumption (G1) satisfies \begin{equation} \label{1.6} \max\{0,{n(n-2)-4\over n(n-2)+4}\}
0$
small in the sequel.\smallskip
Now, we introduce the new functional, which is a truncation of
$I(u)$, as follows: For $u\in H^1_0(\Omega)\backslash\{0\}$,
define
$$
J(u)={1\over2}\int_\Omega|\nabla u|dx-\int_\Omega G(x,u)dx-
\phi(u)\psi(u)\int_\Omega F(x,u)dx,
$$
and $J(0)=0$. The first fact about the functional $J(u)$ is that
$J(u)$ is continuous differentiable.
\begin{prop} \label{prop2.1}
$J\in C^1(H^1_0(\Omega),\mathbb{R})$.
\end{prop}
\begin{proof} Since $I(u)\in C^1(H^1_0(\Omega),\mathbb{R})$,
what we have to prove is that the last term in $J(u)$ is
continuous differentiable at 0 in the sense of Fr\`echet's means.
In fact, by denoting
$$
\mathcal{F}(u)=\phi(u)\psi(u)\int_\Omega F(x,u)dx, \mbox{ for }u\in
H^1_0(\Omega)\backslash\{0\},
$$
it is easy to verify that $0$ is a removable singular point of
$\mathcal{F}(u)$ and by defining $\mathcal{F}(0)=0$ the functional
$\mathcal{F}(u)$ becomes continuous at $0$. Moreover, for any
$\eta\in H^1_0(\Omega)$ and $u\in H^1_0(\Omega)\backslash\{0\}$,
\[
\langle\mathcal{F}'(u),\eta\rangle =
[\langle\phi'(u),\eta\rangle\psi(u)+
\langle\psi'(u),\eta\rangle\phi(u)]\int_\Omega F(x,u)dx
+ \phi(u)\psi(u)\int_\Omega f(x,u)\eta dx.
\]
Then by the assumption (F), the H\"older inequality and (\ref{1.5}), we have
\begin{gather*}
|\langle\psi'(u),\eta\rangle\phi(u)|\int_\Omega|F(x,u)|dx
\leq 2C_3\|u\|^{2^*}_{2^*}|(u,\eta)_{H_0^1}|
\leq 2C_3S^{(1+2^*)/2}\|u\|^{1+2^*}\|\eta\|;\\
\phi(u)\psi(u)\int_\Omega|f(x,u)\eta|dx
\leq C_3\|u\|^{(n+2)/(n-2)}_{2^*}\|\eta\|_{2^*}\\
\leq C_3S^{2^*/2}\|u\|^{(n+2)/(n-2)}\|\eta\|.
\end{gather*}
Thus, we can estimate as follows:
\begin{align*}
&|\langle\phi'(u),\eta\rangle|\phi(u)\int_\Omega|F(x,u)|dx\\
&\leq {2C_3\|u\|^{2^*}_{2^*}\psi(u)\over(2\mu'-1)t}
\Big[|(u,\eta)_{H_0^1}|+\int_\Omega\Big({|g(x,u)|\over\mu}+
2^*A|f(x,u)|\Big)|\eta|dx\Big]\\
&\leq C{\|u\|^{4/(n-2)}\psi(u)\over2\mu'-1}
\Big[\|u\|+\|u\|^p+\|u\|^q+\|u\|^{(n+2)/(n-2)}\Big]\|\eta\|\\
&\leq C\|u\|^{(n+2)/(n-2)}\|\eta\|,
\end{align*}
where $C$ depends only on $S$, $|\Omega|$, $C_1$, $C_2$, $C_3$ and
$\mu'$. Thus we can get the estimate of $\mathcal{F}'$:
$$
\|\mathcal{F}'(u)\|_{H^{-1}(\Omega)}\leq M_1\Big[\|u\|^{1+2^*}+\|u\|
+\|u\|^p+\|u\|^q+\|u\|^{2^*-1}\Big],
$$
which implies
$\mathcal{F}'(u)\to 0$ as $u\to 0$.
With the additional definition $\mathcal{F}'(0)=0$, the above limit
implies that $\mathcal{F}'(u)$ is continuous at $0$.
\end{proof}
\begin{lemma} \label{l.2.2}
If the positive constant $R$ is small enough, for all
$u\in \mathop{\rm supp} \phi \cap \mathop{\rm supp}\psi$ we have
\begin{equation}
|J(u)|\geq M_2\|u\|_{1+p}^{1+p},
\end{equation}
where $M_2$ is a positive constant independent of $u$.
\end{lemma}
\begin{proof} Suppose that $u\in\mathop{\rm supp}\psi=\bar
B_{R}(0)\subset H_0^1(\Omega)$. Then when $R$ is small enough,
from the Sobolev inequality it follows that
\begin{equation*}
2\mu'A\|u\|_{2^*}^{2^*}\leq{1\over2}\|u\|^2.
\end{equation*}
Moreover, if $u\in\mathop{\rm supp}\phi$, by the assumption (G1), we have
\begin{equation} \label{u}
\|u\|^2 \leq 2\mu'\Big({1\over\mu}\int_\Omega G(x,u)dx+
A\|u\|^{2^*}_{2^*}\Big)
\leq 4{\mu'\over\mu}\Big[C_1\|u\|_{1+p}^{1+p}+C_2\|u\|^{1+q}_{1+q}\Big].
\end{equation}
Without loss of generality, we can suppose that $q>p$ and $C_2>0$.
According to the interpolation inequality, we have
\begin{equation}
\|u\|_{1+q}\leq\|u\|^r_{1+p}\|u\|^{1-r}_{2^*},
\end{equation}
where $r=(1+p)(2n-(1+q)(n-2))/(1+q)(2n-(1+p)(n-2))$. Hence we get
\begin{equation} \label{q}
\|u\|_{1+q}^{1+q}\leq S^{1-r\over2}\|u\|_{1+p}^{r(1+p)}
\Big[\|u\|_{1+p}^{1+p}+\|u\|^{1+q}_{1+q}\Big]^{(1-r)(1+q)\over2}\\
\leq C \Big[\|u\|_{1+p}^\alpha+\|u\|^\beta_{1+q}\Big],
\end{equation}
where $C$ is depend only on $S$, and
$\alpha=(1+q)[r+(1-r)(1+p)/2]$ and $\beta=(1+q)[r+(1-r)(1+q)/2]$.
It is clear that $\alpha<\beta$. On the other hand, a simple
calculate shows that $\alpha=(1+p)(n-np+2q+2)/(n-np+2p+2)>1+p$.
Then if $R$ is small enough, (\ref{q}) becomes
$$
\|u\|^{1+q}_{1+q}\leq {1\over2C_2}\|u\|^{1+p}_{1+p}.
$$
Thus we can write (\ref{u}) as
\begin{equation} \label{up}
\|u\|^2\leq8C_1{\mu\over\mu'}\|u\|^{1+p}_{1+p}.
\end{equation}
With (\ref{up}) we can estimate $J(u)$ as follows
\begin{align*}
|J(u)|&\geq \Big(1-{\mu'\over\mu}\Big)\mu\int_\Omega ug(x,u)dx
-(\mu'A+C_3)\|u\|_{2^*}^{2^*}\\
&\geq (\mu-\mu')C_0\|u\|^{1+p}_{1+p}-C\|u\|^{2^*}\\
&\geq (\mu-\mu')C_0\|u\|^{1+p}_{1+p}-C\|u\|^{(1+p)2^*/2}_{1+p}\\
&\geq {1\over2}(\mu-\mu')C_0\|u\|^{1+p}_{1+p},
\end{align*}
where $C$ depends only on $S$, $A$ and $\mu$. Then the lemma
follows with $M_2=(\mu-\mu')C_0/2>0$.
\end{proof}
Although $J(u)$ in generally is not an even functional, $J(u)$
approaches in some sense to such a functional as shown in the
following result.
\begin{prop} \label{prop2.2}
There exists a positive constant $M_3$ independent of $u$, such that
\begin{equation} \label{even}
|J(u)-J(-u)|\leq M_3|J(u)|^\theta,\mbox{ for all }u\in H_0^1(\Omega),
\end{equation}
where $\theta=2^*/2=n/(n-2)$.
\end{prop}
\begin{proof}
From the definition of $J(u)$, the embedding theorem, and
(\ref{up}), we have
\begin{align*}
|J(u)-J(-u)|
&=\phi(u)\psi(u)\big|\int_\Omega F(x,u)dx-\int_\Omega F(-u)dx\big|\\
&\leq 2\phi(u)\psi(u)C_3\|u\|_{2^*}^{2^*}\\
&\leq C\|u\|_{1+p}^{(1+p)2^*/2}\\
&\leq M_3|J(u)|^{2^*/2}.
\end{align*}
The proposition follows with $\theta=2^*/2$. Note that $M_3$ is depend
only on $C_3$ and $S$.
\end{proof}
\section{Verification of Palais-Smale Condition}
Because the functional $I(u)$ contains critical growth nonlinearity, a
well-known fact is that the functional violates Palais-Smale condition.
However, all energy values of the functional where this condition
may fail can be characterized, we refer to Struwe \cite{Struwe}.
The factors $\phi(u)$ and $\psi(u)$ in the new functional $J(u)$ will
change the situation, that is, $J(u)$ remains
most critical points of $I(u)$ and satisfies Palais-Smale condition as
shown in this section.
\begin{lemma} \label{l.3.1}
There exists a suitable constant $R>0$ such
that for any $M>0$, there exists $C(M)>0$, if $|J(u)|\leq M$, then
$$\|u\|^2\leq C(M).$$
\end{lemma}
\begin{proof} From the assumptions on $f(x,u)$ and $g(x,u)$ it
follows that
\begin{equation} \label{M}
|J(u)|\geq{1\over2}\|u\|^2-C\big[\|u\|^{1+p}+\|u\|^{1+q}
+\phi(u)\psi(u)\|u\|^{2^*}\big]
\end{equation}
If $\|u\|^2>R^2$, then $\psi(u)=0$. Without loss of generality, we can suppose
that $\|u\|^2\leq R^2$. Set $R>0$ small enough such that
\begin{equation}
C\phi(u)\psi(u)\|u\|^{2^*}\leq{1\over4}\|u\|^2\quad\mbox{ for all
}u\in H^1_0(\Omega).
\end{equation}
Then (\ref{M}) becomes
\begin{equation}
|J(u)|\geq{1\over4}\|u\|^2-C\Big[\|u\|^{1+p}+\|u\|^{1+q}\Big].
\end{equation}
Therefore, from $|J(u)|\leq M$ it follows that there exists $C(M)$
such that $\|u\|^2\leq C(M)$.
\end{proof}
\begin{prop} \label{PS}
For some suitable positive constants $A$ and $R$, the functional
$J(u)$ satisfies the Palais-Smale condition, that is,
for any sequence $\{u_m\}$ in $H_0^1(\Omega)$ such that $|J(u_m)|\leq M$ and
\begin{equation*}
J'(u_m)\to 0\quad \mbox{as }m\to \infty\quad \mbox{in }H^{-1}(\Omega),
\end{equation*}
then there exists subsequence of $\{u_m\}$ which is convergent in $H_0^1(\Omega)$.
\end{prop}
\begin{proof} Suppose that $\{u_m\}$ is a sequence in
$H_0^1(\Omega)$ with $|J(u_m)|\leq M$ and $J'(u_m)\to 0$ in
$H^{-1}(\Omega)$. What we have to prove is that $\{u_m\}$
possesses a convergent subsequence. Without loss of generality, we
can assume that there exists a positive constant $\epsilon$ such
that $\|u_m\|^2\geq\epsilon$. From lemma \ref{l.3.1} it follows
that the sequence $\{u_m\}$ is bounded in $H_0^1(\Omega)$ since
$|J(u_m)|\leq M$. Thus there exist a subsequence (denoted still by
$\{u_m\}$) and $u$ in $H_0^1(\Omega)$ such that
\begin{gather*}
u_m \rightharpoonup u\quad\mbox{in }H_0^1(\Omega),\\
u_m\to u\quad\mbox{strongly in }L^t(\Omega)\mbox{ for }t\in[1,2^*),\\
u_m\to u\quad\mbox{almost everywhere in }\Omega.
\end{gather*}
Denote
$$
s_m={\|u_m\|^2\over{1\over\mu}\int_\Omega
G(x,u_m)dx+A\|u_m\|^{2^*}_{2^*}}, \quad t_m=\|u_m\|^2.
$$
With this notation, for any $\eta\in C_0^\infty(\Omega)$ we have
\begin{align*}
\langle J'(u_m),\eta\rangle&=
[1-I_1(u_m)]\int_\Omega\nabla u_m\cdot\nabla\eta dx-[1-I_2(u_m)]\int_\Omega g(x,u_m)\eta dx\\
&\quad-\phi(u_m)\psi(u_m)\int_\Omega f(x,u_m)\eta
dx-I_3(u_m)\int_\Omega|u_m|^{4\over n-2}u_m\eta dx,
\end{align*}
where
\begin{align*}
I_1(u_m)&= 2\Big[{\psi(u_m)\Phi'(s_m)\over {1\over\mu}\int_\Omega
G(x,u_m)dx+A\|u_m\|^{2^*}_{2^*}}+\phi(u_m)\Psi'(t_m)
\Big]\int_\Omega F(x,u_m)dx\\
I_2(u_m)&= {\|u_m\|^2\Phi'(s_m)\psi(u_m)\over
\mu\big({1\over\mu}\int_\Omega
G(x,u_m)dx+A\|u_m\|^{2^*}_{2^*}\big)^2}
\int_\Omega F(x,u_m)dx\\
I_3(u_m)&= 2^*A{\|u_m\|^2\Phi'(s_m)\psi(u_m)\over
\mu\big({1\over\mu}\int_\Omega
G(x,u_m)dx+A\|u_m\|^{2^*}_{2^*}\big)^2} \int_\Omega F(x,u_m)dx.
\end{align*}
By Brezis-Lieb's result \cite{BL} (also see (\ref{Lieb1})), we can write
\begin{equation}
\|u_m\|^2=\|u\|^2+\delta'+o(1),\quad
\|u_m\|^{2^*}_{2^*}=\|u\|^{2^*}_{2^*}+\delta+o(1),
\end{equation}
where (subsequence, if necessary)
\begin{equation} \label{Lieb}
\delta'=\lim_{m\to\infty}\|u_m-u\|^2,\mbox{ and }
\delta=\lim_{m\to\infty}\|u_m-u\|^{2^*}_{2^*}.
\end{equation}
It is easy to verify that
\begin{equation} \label{G}
\int_\Omega G(x,u_m)dx\to \int_\Omega G(x,u)dx, \int_\Omega
f(x,u_m)\eta dx\to \int_\Omega f(x,u)\eta dx.
\end{equation}
Denote
$$
\|u\|^2+\delta'=t_0,\quad
{\|u\|^2+\delta'\over{1\over\mu}\int_\Omega
G(x,u)dx+A(\|u\|^{2^*}_{2^*}+\delta)}=s_0.
$$
Moreover (subsequence, if necessary)
$$
\lim_{m\to\infty}\int_\Omega F(x,u_m)dx=r_0.
$$
With this notion, we obviously have
\begin{gather*}
\psi(u_m)=\Psi(t_m)\to \Psi(t_0),\quad \Psi'(t_m)\to \Psi'(t_0),\\
\phi(u_m)=\Phi(s_m)\to \Phi(s_0),\quad \Phi'(s_m)\to \Phi'(s_0).
\end{gather*}
Therefore, as $m$ approaches infinity, we obtain
\begin{align*}
I_1(u_m)&\to I_1=2\Big({\Psi(t_0)\Phi'(s_0)
\over{1\over\mu}\int_\Omega
G(x,u)dx+A(\|u\|^{2^*}_{2^*}+\delta)}+\Phi(s_0)\Psi'(t_0)
\Big)r_0\\
I_2(u_m)&\to I_2=
{(\|u\|^2+\delta')\Phi'(s_0)\Psi(t_0)\over
\mu\big({1\over\mu}\int_\Omega
G(x,u)dx+A(\|u\|^{2^*}_{2^*}+\delta)\big)^2}
r_0\\
I_3(u_m)&\to I_3=
2^*A{(\|u\|^2+\delta')\Phi'(s_0)\Psi(t_0)\over
\mu\big({1\over\mu}\int_\Omega
G(x,u)dx+A(\|u_m\|^{2^*}_{2^*}+\delta)\big)^2} r_0,
\end{align*}
which implies
\begin{equation*}
\langle J'(u_m),\eta\rangle\to\langle\tilde J(u),\eta\rangle,
\end{equation*}
where
\begin{align*}
\langle\tilde J(u),\eta\rangle
&= [1-I_1]\int_\Omega\nabla
u\cdot\nabla\eta dx-[1-I_2]\int_\Omega g(x,u)\eta dx \\
&\quad -\Phi(s_0)\psi(t_0)\int_\Omega f(x,u)\eta
dx-I_3\int_\Omega|u|^{4\over n-2}u\eta dx.
\end{align*}
From $\langle J'(u_m),\eta\rangle=o(1)$, we have
$\langle\tilde J(u),v\rangle=0$ for all $v$ in $H_0^1(\Omega)$. It follows
that
\begin{equation} \label{<>}
\langle J'(u_m)-\tilde J(u),u_m-u\rangle=
\langle J'(u_m),u_m-u\rangle\to 0.
\end{equation}
On the other hand, we have
\begin{equation} \label{<.>}
\begin{aligned}
&\langle J'(u_m)-\tilde J(u),u_m-u\rangle\\
&=(1-I_1)\|u_m-u\|^2+o(1)
-(1-I_2)\int_\Omega\Big[g(x,u_m)-g(x,u)\Big](u_m-u)dx\\
&\quad -\Psi(t_0)\Phi(s_0)\int_\Omega\Big[f(x,u_m)-f(x,u)\Big](u_m-u)dx\\
&\quad -I_3\int_\Omega\Big[|u_m|^{4\over n-2}u_m-|u|^{4\over
n-2}u\Big](u_m-u)dx\\
&= (1-I_1)\|u_m-u\|^2+o(1) -\Psi(t_0)\Phi(s_0)\int_\Omega f(x,u_m-u)(u_m-u)dx\\
&\quad -I_3\int_\Omega|u_m-u|^{2^*}dx.
\end{aligned}
\end{equation}
where we have used the fact
$\int_\Omega[f(x,u_m)-f(x,u)](u_m-u)dx$=$\int_\Omega
f(x,u_m-u)(u_m-u)dx+o(1)$, which proof can be founded in the next
lemma. Before proceeding furthermore, we first claim that
$|I_1|\leq1/2$ and $|I_3|\leq1/4$. In fact, if
$u_m\not\in\mathop{\rm supp}\phi\cap\mathop{\rm supp}\psi$,
then $I_1(u_m)=I_3(u_m)=0$,
we are done. In the contrary case, that is,
$u_m\in\mathop{\rm supp}\phi\cap\mathop{\rm supp}\psi$, we can suppose that
$u\in\mathop{\rm supp}\Psi'$. Otherwise, we have $I_3(u_m)=0$ and
$I_1(u_m)=\phi(u_m)\Psi'(t_m)\int_\Omega F(x,u_m)dx$, the desired
result easy follows. Without loss generality, we can suppose that
$t_m\in\Psi'$, namely
\begin{gather*}
1 \leq \|u_m\|^2\Big({1\over\mu}\int_\Omega G(x,u_m)dx+A\|u_m\|_{1+p}^{1+p}
\Big)^{-1}\leq2\mu',\\
{R^2\over2} \leq \|u_m\|^2\leq R^2.
\end{gather*}
By the choice of $\Phi$ and $\Psi$, we can estimate $I_1(u_m)$ as
\begin{align*}
|I_1(u_m)|&\leq \Big[{4C_3(2\mu'-1)^{-1}\over{1\over\mu}\int_\Omega
G(x,u_m)dx+A\|u_m\|^{2^*}_{2^*}}
+{4C_3\over R^2}\Big]\|u_m\|^{2^*}_{2^*}\\
&\leq 4C_3S^{-{2^*\over2}}\big({2\mu'\over2\mu'-1}+1\big)\|u\|^{4\over n-2}\\
&\leq {4C_3S^{n\over2-n}\over2\mu'-1}R^{4\over n-2}.
\end{align*}
Let $R$ be small enough, then $|I_1(u_m)|<1/2$, which implies that
$|I_1|\leq1/2$.
In a similar way, we can estimate $I_3$ as
\begin{align*}
|I_3(u_m)|&\leq 4{2^*AC_3\mu'\over2\mu'-1} \Big(\int_\Omega
G(x,u_m)dx+A\|u_m\|^{2^*}_{2^*}\Big)^{-1} \|u_m\|^{2^*}_{2^*}\\
&\leq 2^*AC_3S^{n\over2-n}{8\mu'^2\over2\mu'-1}\|u\|^{n\over n-2}\\
&\leq 2^*AC_3S^{n\over2-n}{8\mu'^2\over2\mu'-1}R^{n\over n-2}.
\end{align*}
After the constant $A$ being fixed, we can set $R$ be small enough such
that $|I_3(u_m)|<1/4$, which implies that $|I_3|\leq1/4$.
Let us go back to (\ref{<>}) and (\ref{<.>}).
If $u_m\in B_{R}(0)=\mathop{\rm supp}\psi$
and for $R$ small enough
\begin{align*}
o(1)&\geq {1\over2}\|u_m-u\|^2
-\Psi(t_0)\Phi(s_0)S^{n\over2-n}\|u_m-u\|^{2^*}
-{1\over4}\|u_m-u\|^{2^*}\\
&\geq {1\over4}\|u_m-u\|^2.
\end{align*}
Where we have used the fact
$$
\|u_m-u\|^{2^*}\leq(\|u_m\|^{n\over n-2}+\|u\|^{n\over
n-2})\|u_m-u\|^2 \leq2R^{n\over n-2}\|u_m-u\|^2.
$$
If $u_m\not\in\mathop{\rm supp}\psi$, we still have
$$
o(1)\geq{1\over2}\|u_m-u\|^2.
$$
As a consequence, for a subsequence,
$u_m\to u\quad\mbox{strongly in }H_0^1(\Omega)$.
\end{proof}
\begin{lemma} \label{h(u)}
Suppose that $h(x,u)\in$$C^1(\Omega\times(-\infty,+\infty))$ and
$|h'_u(x,u)|\leq C(1+|u|^{4/(n-2)})$. If $u_m\rightharpoonup u$
weakly in $H^1_0(\Omega)$, then
\begin{equation} \label{Lieb1}
\int_\Omega[h(x,u_m)-h(x,u)](u_m-u)dx=\int_\Omega
h(x,u_m-u)(u_m-u)dx+o(1)
\end{equation}
\end{lemma}
\begin{proof}
The hypothesis on the growth of $h$ implies that
$h(x,u_m)$ and $h'(x,u_m)u_m$ are bounded in
$L^{2n/(n+2)}(\Omega)$. The compact embedding theorem for
Sobolev spaces yields that for a subsequence,
\begin{equation} \label{'}
h(x,u_m)\to h(x,u),\,\, h'_u(x,u_m)u_m\to
h'_u(x,u)u\quad\mbox{strongly in }L^{2n\over n+2}(\Omega).
\end{equation}
Furthermore we can deduce that
$$
\int_\Omega[h(x,u_m)-h(x,u)]udx=o(1),\quad\mbox{and}\quad \int_\Omega
h(x,u)(u_m-u)dx=o(1).
$$
Consequently
$$
\int_\Omega[h(x,u_m)-h(x,u)](u_m-u)dx=\int_\Omega[h(x,u_m)u_m-h(x,u)u]dx+o(1).
$$
On the other hand, as $n\to\infty$,
\begin{align*}
&\int_\Omega\Big[h(x,u_m)u_m-h(x,u_m-u)(u_m-u)\Big]dx\\
&= \int_\Omega\int_0^1{d\over dt}\left\{h(x,u_m-(1-t)u)(u_m-(1-t)u)\right\}dt\,dx\\
&= \int_\Omega\int_0^1\left[h'_u(x,u_m-(1-t)u)(u_m-(1-t)u)u
+h(x,u_m-(1-t)u)u\right]dt\,dx\\
&= \int_0^1\int_\Omega\left[h'_u(x,u_m-(1-t)u)(u_m-(1-t)u)u
+h(x,u_m-(1-t)u)u\right]dx\,dt\\
&\to \int_0^1\int_\Omega[h'_u(x,tu)tu^2+h(x,tu)u]dt\,dx\\
&= \int_0^1\int_\Omega{d\over dt}\left\{h(x,tu)(tu)\right\}dt\,dx
=\int_\Omega h(x,u)u\,dx,
\end{align*}
where the weak convergence limit is a consequence of (\ref{'}).
\end{proof}
\noindent\textbf{Remark.} If $f$ is convex, we can infer that the first
inequality in (F) implies the second one. Moreover, a revised
proof as in Brezis-Lieb's paper \cite{BL} can be used to establish
that differentiability of $f$ is not necessary.
\section{The Construction of Critical Values}
In this section, we establish a series of minimax sequences of the
functional $J(u)$ and prove at last that there is a subsequence of
them which is the infinitely many critical values of the
functional $I(u)$.
Denote the eigenvalue of $-\Delta$ with vanish boundary value by
$\lambda_k$, $k=1,2,\dots$, and the normalized eigenfunction
corresponding to $\lambda_k$ by $e_k$. Set
\begin{gather*}
E_k= \mathop{\rm span}\{e_1,e_2,\dots,e_k\},\\
S_k= \{u\in E_k;\|u\|=1\}\\
S_{k+1}^+= \{u=te_{k+1}+w;\|u\|=1,w\in E_k,t\geq0\}.
\end{gather*}
Define the map sets as follows:
\begin{gather*}
\Lambda_k= \{h\in C(S_k,H_0^1(\Omega));h\mbox{ is odd map}\},\\
\Gamma_k= \{h\in
C(S^+_{k+1},H_0^1(\Omega));h|_{S_k}\in\Lambda_k\}.
\end{gather*}
With these sets of maps, we can define minimax sequence of $J(u)$
as follows:
$$
b_k=\inf_{h\in\Lambda_k}\max_{u\in S_k}J(h(u)),\quad
c_k=\inf_{h\in\Gamma_k}\max_{u\in S_{k+1}^+}J(h(u)).
$$
For $\delta>0$, we set
\begin{gather*}
\Gamma_k(\delta)= \{h\in\Gamma_k;J(h(u))\leq b_k+\delta,u\in S_k\}\\
c_k(\delta)= \inf_{h\in\Gamma_k(\delta)}\max_{u\in S_{k+1}^+}J(h(u)).
\end{gather*}
It is easy to prove that the above notation are well-defined and
$b_k\leq c_k\leq b_{k+1}$, $c_k\leq c_k(\delta)$.
From the definition of $J(u)$ and the assumptions (G1) and (F), it
follows that, if $R>0$ is small,
$$
J(u)\leq{1\over2}\|u\|^2-C_0\|u\|_{p+1}^{p+1}+\phi(u)\psi(u)C_3\|u\|^{2^*}_{2^*}\leq\|u\|^2-C_0\|u\|_p^p.
$$
By setting $H(u)=\rho u$, we obviously have $H\in\Lambda_k$. Since
$H(S_k)\subset E_k$, we can find out $\rho=\rho_k$ and $C_k>0$
such that for any $u\in S_k$,
\begin{equation} \label{b_k<0}
J(H(u))=J(\rho_ku)\leq \rho_k^2-C_0C_k\rho_k^p<0,
\end{equation}
which implies that $J(u)<0$ for all $u$ in $(B_{\rho_k}(0)\cap
E_k)\backslash\{0\}$ and $b_k<0$ for $k=1,2,\dots$. Furthermore,
for any $\delta>0$, it is clear that $c_k(\delta')\leq
c_k+\delta$, where $\delta'=c_k-b_k+\delta$. However, an important
fact is that $c_k(\delta)<0$ for each $k$ and each $\delta>0$.
Before giving the proof of the fact, we first claim that the
functional $J(u)$ possesses no critical point with nonnegative
critical value except the origin. In fact, suppose $u\in K=\{u\in
H_0^1(\Omega);J'(u)=0\}$ and $J(u)\geq0$, then
\begin{align*}
0&\leq J(u)=J(u)-{1\over2}\langle J'(u),u\rangle\\
&\leq ({1\over2}-\mu)\int_\Omega g(x,u)udx+
\phi(u)\psi(u)\Big|\int_\Omega\Big(F(x,u)+f(x,u)u\Big)dx\Big|\\
&\quad+\big[
|\langle\phi'(u),u\rangle|\psi(u)+|\langle\psi'(u),u\rangle|\phi(u)\big]
\int_\Omega|F(x,u)|dx\\
&\leq ({1\over2}-\mu)C_0\|u\|^{1+p}_{1+p}
+C_3\big[\phi(u)\psi(u)+M_0\psi(u)+8\phi(u)\big]\|u\|_{2^*}^{2^*}\\
&\leq {1\over2}({1\over2}-\mu)C_0\|u\|^{1+p}_{1+p}\leq0,
\end{align*}
for suitable small $R$, which leads to $u=0$.
\begin{lemma} \label{l.1.3}
For $k=1,2,\dots,$ and $\delta>0$ we have
$c_k(\delta)<0$.
\end{lemma}
\begin{proof} Without loss of generality, we suppose that
$b_k+\delta<0$. From the definition of $c_k(\delta)$ it follows
that there exits $h\in\Lambda_k$ such that
\begin{equation} \label{2.1}
\max_{u\in S_k}J(h(u))\leq b_k+{\delta\over2}.
\end{equation}
Denote the orthogonal projective operator from $H_0^1(\Omega)$ to
$E_m$ by $P_m$. Since $h(S_k)$ is a compact set in
$H_0^1(\Omega)$, it is not difficult to show that there exists a
positive integer $m$ ($m\geq k$) such that
\begin{equation} \label{2.2}
\max_{u\in S_k}J(P_mh(u))\leq b_k+\delta.
\end{equation}
Note that $P_mh\in\Lambda_k$. The fact that the functional
$J(u)$ possesses no critical point with nonnegative critical value
except the origin implies that $B_{\rho/2}(0)\subset
H_0^1(\Omega)$ is a neighborhood of $K_0$, where $K_a=\{u\in
H_0^1(\Omega);J(u)=a,J'(u)=0\}$. Let $\bar\epsilon=-(b_k+\delta)/2$ and
$\rho=\min\{\rho_{m+1},\mbox{dist}(0,J_{-\bar\epsilon})\}$, where
$\rho_{m+1}$ is determined as in (\ref{b_k<0}). Therefore, the
deformation theorem can be used to find out a positive
$\epsilon\in(0,\bar\epsilon)$ and a continuous map $\eta\in
C(H_0^1(\Omega)\times[0,1],H_0^1(\Omega))$, such that
\begin{itemize}
\item[(i)] $\eta(u,1)=u$, for all $u\not\in
J^{-1}(-\bar\epsilon,\bar\epsilon)$
\item[(ii)] $\eta(J_\epsilon\backslash B_{\rho/2}(0),1)\subset J_{-\epsilon}$.
\end{itemize}
The idea in the following is to seek a contractible subset of
$J_\epsilon$ and to expend the map $P_mh$ in the subset before it can be
deformed into $J_{-\epsilon}$ with the deformation map $\eta$.
Indeed, in view of $P_mh(S_k)\subset E_m$, it is natural for us to
consider the functional $\tilde J$, the restriction of $J$ on
$E_m$, namely $\tilde J=J|_{E_m}$. It is clear that $\tilde J\in
C^1(E_m,\mathbb{R})$ and by the same argument as previous shown, one
can obtain that $\tilde J$ possesses no critical point with
nonnegative critical value except the origin. Thus that fact
implies that the level set $\tilde J_\sigma=\{u\in E_m;\tilde
J(u)<\sigma\}$ is a deformation retract of $E_m$ for any
$\sigma>0$, so is $\tilde J_\epsilon$ contractible, for the positive
$\epsilon$ found in the previous deformation theorem. Hence the map
$P_mh$ can be extended as
\begin{equation*}
\widetilde{P_mh}:S_{k+1}^+\to\tilde J_\epsilon.
\end{equation*}
Let $T$ be a map from $E_m$ to $E_{m+1}$ defined by
$$
T(u)=\begin{cases}
u,&u\not\in\bar B_\rho(0)\cap E_m\\
u+\sqrt{\rho^2-\|u\|^2}e_{m+1},&u\in\bar B_\rho(0)\cap E_m.
\end{cases}
$$
It is clear that $T$ is continuous, and
$$
(T\circ\widetilde{P_mh})[S_{k+1}^+]\cap B_{\rho/2}(0)=\emptyset.
$$
Since $\widetilde{P_mh}[S_{k+1}^+]\subset\tilde J_\epsilon\subset J_\epsilon$,
we also have $T(\widetilde{P_mh}[S_{k+1}^+])\subset J_\epsilon$.
Denote $H(\cdot)=\eta(T\circ\widetilde{P_mh}(\cdot),1)$ and
$A=H(S_{k+1}^+)$. From (i) it follows that
$H|_{S_{k+1}^+}=P_mh\in\Lambda_k$ which implies that
$H\in\Gamma_k(\delta)$. Moreover from (ii) it follows that
$A=H(S_{k+1}^+)\subset J_{-\epsilon}$. Therefore,
$$
\max_{u\in A}J(u)\leq-\epsilon<0
$$
which implies that $c_k(\delta)<0$.
\end{proof}
\begin{lemma} \label{l.1.4}
Set $d_k=|b_k|$. Then there exists a positive constant $M_4$ such
that for all $k$ large enough
\begin{equation}
d_k\leq M_4k^{-{2\over n}{1+p\over1-p}},
\end{equation}
where $M_4$ is independent of $k$.
\end{lemma}
\begin{proof} For any fixed $h\in\Lambda_k$ ($k\geq2$), from
Borsuk-Ulam Theorem it follows that
\[
h(S_k)\cap E_{k-1}^\perp\neq \emptyset.
\]
Take $w\in h(S_k)$, denote $\rho=\|w\|$. In $E_{k-1}^\perp$, we have
\[
\|u\|^2\geq\lambda_k\|u\|_2^2.
\]
With the properties of $\phi(u)$ and $\psi(u)$, we can estimate $J(w)$ as
follows:
\begin{align*}
J(w)&\geq {1\over2}\|w\|^2-C_1\|w\|^{1+p}_{1+p}-C_2\|w\|^{1+q}_{1+q}
-C_3\phi(w)\psi(w)\|w\|^{2^*}_{2^*}\\
&\geq {1\over4}\|w\|^2-C\|w\|^{1+p}_2\\
&\geq {1\over4}\rho^2-C\lambda_k^{-{1+p\over2}}\rho^{1+p}
=Q(\rho).
\end{align*}
Hence
\[
\max_{u\in h(S_k)}J(u)\geq J(w)
\geq \inf_{u\in\partial B_\rho(0)\cap E_{k-1}^\perp}J(u)
\geq \inf_{\rho\geq0}Q(\rho)\geq-M_5\lambda_k^{-{1+p\over1-p}}.
\]
On the other hand, $\lambda_k\geq M_6k^{2/n}$; see for example \cite{R}).
By the arbitrariness of $h$, we get
$b_k\geq-M_4k^{2(1+p)/n(p-1)}$.
\end{proof}
\begin{lemma} \label{l.1.5}
If $c_k=b_k$ for all $k$ large enough, then
\[
d_k\geq M_7k^{-{1\over\theta-1}},
\]
where $\theta=2^*/2=n/(n-2)$ and $1/(\theta-1)=(n-2)/2$ and $M_7$
is independent of $k$.
\end{lemma}
\begin{proof} Suppose that for some $k_0$, we have $b_k=c_k$
($k=k_0,k_0+1,\dots$). Then for any $\epsilon\in(0,-b_k)$, there exists a
map $H\in\Gamma_k$ such that
\begin{equation}
b_k\leq J(H(u_0))=\max_{u\in S^+_{k+1}}J(H(u))\leq c_k+\epsilon=b_k+\epsilon.
\end{equation}
Define $\bar H:S_{k+1}\to H_0^1(\Omega)$ as
$$
\bar H=\begin{cases}
H(u),&u\in S_{k+1}^+\\-H(-u),&-u\in S_{k+1}^+.
\end{cases}
$$
Since $S_{k+1}=S^+_{k+1}\cup(-S^+_{k+1})$, it is clear that
$\bar H\in\Lambda_{k+1}$, that is,
\begin{equation} \label{max}
b_{k+1}\leq \max_{u\in S_{k+1}}J(\bar H(u))
= \max\big\{\max_{u\in S^+_{k+1}}J(H(u)),\max_{-u\in S^+_{k+1}}J(-H(-u))\big\}.
\end{equation}
We claim that
\begin{equation} \label{b_k}
b_{k+1}\leq b_k+\epsilon+M_3|b_{k+1}|^\theta.
\end{equation}
In fact, if
$$
\max_{u\in S^+_{k+1}}J(H(u))\geq\max_{-u\in S^+_{k+1}}J(-H(-u)),
$$
then (\ref{max}) implies $b_{k+1}\leq b_k+\epsilon$ which leads to (\ref{b_k}).
On the contrary, we can use Lemma \ref{even}
\begin{equation} \label{J(-v)}
J(-v)\leq J(v)+M_3|J(-v)|^\theta,\mbox{ for all }v\in H_0^1(\Omega),
\end{equation}
where $\theta=2^*/2$. Let $v_0$ be in $H(S_{k+1}^+)$ such that
$$
J(-v_0)=\max_{v\in H(S_{k+1}^+)}J(-v),
$$
then (\ref{J(-v)}) becomes $J(-v_0)\leq b_k+\epsilon+M_3|J(-v_0)|^\theta$, or
\begin{equation}
b_{k+1}\leq b_k+\epsilon+M_3\big|\max_{v\in H(S^+_{k+1})}J(-v)\big|^\theta.
\end{equation}
If $\max_{v\in H(S^+_{k+1})}J(-v)\leq0$, then (\ref{b_k}) can be
drawn directly from the above inequality. Otherwise, we can take
some $v=v_0\in H(S_{k+1}^+)$ such that
$$
b_{k+1}\leq J(-v_0)\leq0,
$$
we still get (\ref{b_k}) in the same way. Now by letting $\epsilon\to 0$ we
get
\begin{equation} \label{b_kk}
b_{k+1}\leq b_k+M_3|b_{k+1}|^\theta,\mbox{ for }k\geq k_0.
\end{equation}
With induction, we can show the lemma from (\ref{b_kk}), we omit the details.
\end{proof}
\section{Proof of The Main Theorems}
\begin{lemma} \label{lem5.1}
If $c_k>b_k$, then for $\delta\in(0,c_k-b_k)$, $c_k(\delta)$ is a critical
value of the functional $J(u)$.
\end{lemma}
\begin{proof} If this is not the case, that is, $c_k>b_k$, but for
$\delta\in(0,c_k-b_k)$, $c_k(\delta)$ is not a critical value of
the functional $J(u)$. Set $\bar\epsilon=c_k-b_k-\delta$. Then
$\bar\epsilon>0$. From the deformation theorem it follows that there
exist a positive constant $\epsilon\in(0,\bar\epsilon)$ and a continuous map
$\eta(\cdot,\cdot)\in C(H_0^1(\Omega)\times[0,1],H_0^1(\Omega))$
such that
\begin{itemize}
\item[(i)] $\eta(u,t)=u$, for all $u\not\in J^{-1}(c_k(\delta)
-\bar\epsilon,c_k(\delta)+\bar\epsilon)$
\item[(ii)] $\eta(J_{c_k(\delta)+\epsilon},1)\subset J_{c_k(\delta)-\epsilon}$.
\end{itemize}
By the definition of $c_k(\delta)$, there exists
$H\in\Gamma_k(\delta)$ such that
\begin{equation}
\max_{u\in S_{k+1}^+}J(H(u))\leq c_k(\delta)+\epsilon.
\end{equation}
For $u$ in $S_k$, we also have $J(H(u))\leq
b_k+\delta=c_k-\bar\epsilon\leq c_k(\delta)-\bar\epsilon$. Set
$$
\bar H(u)=\eta(H(u),1),\mbox{ for all }u\in S_k.
$$
From (i) it follows that $\bar H\in\Gamma_k(\delta)$, but (ii)
implies that $\bar H(J_{c_k(\delta)+\epsilon})\subset
J_{c_k(\delta)-\epsilon}$, which arises a contradiction.
\end{proof}
\begin{lemma} \label{l.5.4}
Suppose that $1/(\theta-1)=(n-2)/2>2(1+p)/(1-p)n$. Then there exists a
subsequence $c_{k_j}$ of $c_k$ ($j=1,2,\dots$) such that
\begin{equation} \label{c>b}
c_{k_j}>b_{k_j},j=1,2,\dots.
\end{equation}
\end{lemma}
\begin{proof} With Lemma \ref{l.1.4} and Lemma \ref{l.1.5}, if for
all $k$ large enough $c_k=b_k$, then we have
\begin{equation} \label{<}
M_7k^{-{1\over\theta-1}}\leq d_k\leq M_4k^{-{2\over n}{1+p\over1-p}}.
\end{equation}
Since $1/(\theta-1)<2(1+p)/n(1-p)$, the above inequality will lead to a contradiction for large $k$.
Hence the subsequence $c_{k_j}$ satisfying (\ref{c>b}) must exist.
\end{proof}
\begin{lemma} \label{l.5.3}
Suppose that $p$ satisfies
$p>\max\{0,(n(n-2)-4)/(n(n-2)+4)\}$. Let $u_{k_j}$ be the critical
points of $J(u)$ corresponding to the critical values
$c_{k_j}(\delta_j)$, where $\delta_j=(c_{k_j}-b_{k_j})/2$. Then we
have
\[
\|u_{k_j}\|\to 0\quad \mbox{as }j\to \infty.
\]
\end{lemma}
\begin{proof} By the previous lemmas, for the subsequence
$\{c_{k_j}\}_{j=1}^\infty$, we have
\begin{equation} \label{last}
-M_4k_j^{-{2\over n}{1+p\over1-p}} \leq b_{k_j}