\documentclass[reqno]{amsart}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 100, pp. 1--18.\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-2004/100\hfil Strongly indefinite functionals]
{Strongly indefinite functionals with perturbed symmetries and
multiple solutions of nonsymmetric elliptic systems}

\author[M. Clapp, Y. Ding, S. Hern\'{a}ndez-L.\hfil EJDE-2004/100\hfilneg]
{M\'{o}nica Clapp, Yanheng Ding, Sergio Hern\'{a}ndez-Linares} % in alphabetical order

\address{M\'{o}nica Clapp\hfill\break
Instituto de Matem\'{a}ticas\\
Universidad Nacional Aut\'{o}noma de M\'{e}xico\\
Circuito Exterior, Ciudad Universitaria\\
04510 M\'{e}xico D.F., Mexico}
\email{mclapp@math.unam.mx}

\address{Yanheng Ding \hfill\break
Institute of Mathematics, AMSS\\
Chinese Academy of Sciences\\
100080 Beijing, China}
\email{dingyh@math.ac.cn}

\address{Sergio Hern\'{a}ndez-Linares \hfill\break
Departamento de Matem\'{a}ticas\\
Universidad Aut\'{o}noma Metropolitana - Iztapalapa\\
Av. Michoac\'{a}n y la Pur\'{\i}sima, Col. Vicentina\\
09340 M\'{e}xico D.F., Mexico}
\email{slinares@math.unam.mx}

\date{}
\thanks{Submitted March 12, 2004. Published August 18, 2004.}
\thanks{M. Clapp was supported by grant IN110902-3 from PAPIIT, UNAM. \hfill\break\indent
Y. Ding  was supported by the ``973'' project and NSFC 19971091 of China.\hfill\break\indent
S. Hernandez was supported by grant IN110902-3 from PAPIIT, UNAM}
\subjclass[2000]{35J50, 58E05}
\keywords{Critical point theory; perturbation of symmetries; elliptic systems;
\hfill\break\indent
strongly indefinite functionals;  multiple solutions; critical Sobolev exponent}

\begin{abstract}
 We prove a critical-point result which provides conditions for 
 the existence of infinitely many critical points of a strongly 
 indefinite functional with perturbed symmetries. Then we apply 
 this result to obtain infinitely many solutions of non-symmetric 
 super-quadratic noncooperative elliptic systems, allowing some 
 supercritical growth.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

Consider the noncooperative elliptic system
\begin{equation} \label{P}
\begin{gathered}
-\Delta u=| u| ^{p-2}u+f_{u}(x,u,v) \quad \hbox{in }\Omega \\
\Delta v=| v| ^{q-2}v+f_{v}(x,u,v) \quad \hbox{in } \Omega \\
u=0,\quad v=0 \quad \hbox{on }\partial \Omega\,,
\end{gathered}
\end{equation}
where $\Omega \subset \mathbb{R}^{N}$ is a bounded smooth domain, $N\geq 3$,
$p\in (2,2^{\ast })$, $q\in [ 2,\infty )$, and $f\in C^{1}(\overline{%
\Omega }\times \mathbb{R}^2,\mathbb{R})$ is a lower order term, which is
not necessarily symmetric in $(u,v)$. As usual, $2^{\ast }:=\frac{2N}{N-2}$
denotes the critical Sobolev exponent.

In the previous decades there has been a great amount of activity in the study
of elliptic systems. Elliptic systems leading to strongly indefinite
functionals have been studied, for example, in \cite{bc, bd, br, cm1, cm2,
df, dm, hv, ks}. However, only subcritical systems have been considered in
these papers, and the multiplicity results therein require some symmetry
assumption on $f$. Recently De Figueiredo and Ding \cite{dd} considered the
case $q\geq 2^{\ast }$. Under appropriate growth conditions, they
established the existence of infinitely many solutions of $\mathbf{(}\wp
\mathbf{)}$ for functions $f$ which are even in $(u,v)$. Here we will show
that one can do without the symmetry assumption. Namely, we prove the
following.
\begin{theorem}\label{intro}
If $p\in (2,\frac{2N-2}{N-2})$, $q\in [ p,\infty )$, and $%
f\in C^{1}(\overline{\Omega }\times \mathbb{R}^2,\mathbb{R})$ satisfies
\begin{gather*}
| f_{u}(x,u,v)| \leq c(| u| ^{\gamma p-1}+|
v| ^{\sigma -1}+1) \\
| f_{v}(x,u,v)| \leq c(| u| ^{\gamma p-1}+|
v| ^{\gamma q-1}+1)
\end{gather*}
for all $(x,u,v)\in \Omega \times \mathbb{R}^2$, and some $c>0$, $0\leq
\sigma -1\leq \frac{q}{p}(\gamma p-1)$, and $\frac{1}{p}\leq \gamma <%
\min \{( \frac{1}{p}-\frac{1}{2^{\ast }}) N,\frac{q}{q-1}(\frac{2^*-1}{2^*})\}$,
 then \eqref{P} has infinitely many solutions.
\end{theorem}

This result is a special case of a stronger result (Theorem \ref{homo})
which is obtained as an application of an abstract critical point theorem
for strongly indefinite functionals with perturbed symmetries which we state
and prove in section 2.

Variational methods for establishing existence of infinitely many solutions
of an elliptic equation with perturbed symmetries were first introduced by
Bahri and Berestycki \cite{bb}, Struwe \cite{s} and Rabinowitz \cite{r1} in
the early eighties, further developed by Bahri and Lions \cite{bl} and
Tanaka \cite{t} and, more recently, by Bolle, Ghoussoub and Tehrani
\cite{bo, bgt}, among others.

On the other hand, various methods for dealing with symmetric strongly
indefinite functionals are now well known. The first one is due to
Rabinowitz \cite{r2} who reduced the indefinite problem to a
finite-dimensional one. Another useful approach is due to Benci and
Rabinowitz \cite{br} who showed that the original methods of critical point
theory still work if one restricts the class of deformations appropriately.
A different approach, based on a Galerkin type approximation, was given by
Bartsch and Clapp in \cite{bc}.

The abstract result we present here is also based on a Galerkin type
approximation which reduces the study of strongly indefinite functionals
with perturbed symmetries to a semidefinite situation, thus allowing the use
of Morse theory methods as in \cite{bl} and \cite{t}. However, unlike the
symmetric case or the semidefinite case, the strongly indefinite perturbed
case requires fine knowledge on the topology of the sublevel sets of the
approximations of the associated symmetric functional (see Remark \textsl{%
(e) } at the end of section 2). The key step in the proof of Theorem \ref%
{intro} consists in a careful study of such sublevel sets. Our description
will yield, in addition, estimates for the energy of the solutions of the
system $(\wp )$, similar to those given by Bahri and Lions \cite{bl} in the
symmetric single equation case, and recently extended by Castro and Clapp
\cite{cc} to the perturbed single equation case (see Theorem \ref{homo}).

This paper is organized as follows. In section 2 we state and prove an
abstract critical point result for strongly indefinite functionals with
perturbed symmetries, and in section 3 we apply this result to prove Theorem %
\ref{intro}.

\section{Strongly indefinite functionals with perturbed symmetries}

Let $X$ be a Banach space with a direct sum decomposition $X=X^{+}\oplus
X^{\ast }\oplus X^{-}$. According to this decomposition, a point in $X$ will
be denoted $u=(u^{+},u^{\ast },u^{-})$. Let
\begin{equation*}
X_{1}^{+}\subset X_{2}^{+}\subset \dots \subset X^{+}, \quad
X_{1}^{\ast }\subset X_{2}^{\ast }\subset \dots \subset X^{\ast },\quad
X_{1}^{-}\subset X_{2}^{-}\subset \dots \subset X^{-}
\end{equation*}
be sequences of finite dimensional linear subspaces of $X^{+}$, $X^{\ast }$
and $X^{-}$ such that $\dim X_{k}^{+}=k$. For $k,n\geq 1$ we write
\begin{equation*}
X^n:=X^{+}\oplus X_{n}^{\ast }\oplus X_{n}^{-}\quad \hbox{and}\quad
X_{k}^n:=X_{k}^{+}\oplus X_{n}^{\ast }\oplus X_{n}^{-}.
\end{equation*}
Let $\iota :X\to X$ be the involution
\begin{equation*}
\iota (u^{+},u^{\ast },u^{-})=(-u^{+},u^{\ast },-u^{-}).
\end{equation*}
Then $X^{\ast }=\{u\in X:\iota u=u\}$ is the fixed point set of$\ \iota $.
We say that a subspace $V$ of $X$ is $\iota $-invariant if $\iota u\in V$
for every $u\in V$, and we say that a map $\sigma :V\to W$ between
two $\iota $-invariant subspaces is $\iota $-equivariant if $\sigma (\iota
u)=\iota (\sigma (u))$ for every $u\in V$.

Let $\Phi :X\times [ 0,1]\to \mathbb{R}$ be a $C^{1}$-functional,
and let $\Phi ^n:X^n\times [ 0,1]\to \mathbb{R}$ be its restriction to
$X^n\times [ 0,1]$. We think of $\Phi $ and
$\Phi ^n $ as being paths of functionals
\begin{gather*}
\Phi _{t} :X\to \mathbb{R},\quad \Phi _{t}(u)=\Phi (u,t),
\quad 0\leq t\leq 1, \\
\Phi _{t}^n :X^n\to \mathbb{R},\quad \Phi_{t}^n(u)=\Phi ^n(u,t),
\quad 0\leq t\leq 1,\hbox{ }n\geq 1,
\end{gather*}
and write
\begin{equation*}
\Phi _{t}'(u):=\frac{\partial }{\partial u}\Phi (u,t),\quad
(\Phi _{t}^n)'(u):=\frac{\partial }{\partial u}\Phi ^n(u,t).
\end{equation*}
for their derivatives with respect to $u$.  We assume that $\Phi $
satisfies the following assumptions.

\begin{itemize}
\item[(H1)] Every sequence $(u_{k},t_{k})$ in $X\times [ 0,1]
$ with $u_{k}\in X^{n_{k}}$, $n_{k}\to \infty $, $t_{k}\to
t, $ $\Phi _{t_{k}}(u_{k})\to c$, $\| (\Phi
_{t_{k}}^{n_{k}})'(u_{k})\| \to 0$, has a subsequence
converging in $X$ to a critical point of $\Phi _{t}$.

\item[(H2)] For every $n\in \mathbb{N}$ large enough and $b\in
\mathbb{R}$ there is a constant $C=C(n,b)$ such that
\begin{equation*}
\quad \big| \frac{\partial }{\partial t}\Phi (u,t)\big|
\leq C(\|(\Phi _{t}^n)'(u)\| +1)(\|
u\| +1)\quad \hbox{if  } u\in X^n ,\;  | (\Phi_{t}^n)(u)| \leq b.
\end{equation*}

\item[(H3)] There exist two continuous functions $\theta
_{1},\theta _{2}:[0,1]\times \mathbb{R}\to \mathbb{R}$, $\theta
_{1}\leq \theta _{2}$, which are Lipschitz continuous in the second variable
and such that
\begin{equation*}
\theta _{1}(t,\Phi _{t}(u))\leq \frac{\partial }{\partial t}\Phi (u,t)\leq
\theta _{2}(t,\Phi _{t}(u))\quad \hbox{if }\Phi _{t}'(u)=0.
\end{equation*}

\item[(H4)] For every finite dimensional subspace $W$ of $X$ and $%
a\in \mathbb{R}$ there exists an $R>0$ such that $\Phi _{t}(w)\leq a$ for
every $t\in [ 0,1]$, $w\in W$ with $\| w\| \geq R$.

\item[(H5)] $\Phi _{0}(\iota u)=\Phi _{0}(u)$ for every $u\in X$,
and there exists an $M\geq 0$ such that $\Phi _{t}(u^{\ast })\leq M$ for
every $t\in [ 0,1]$, $u^{\ast }\in X^{\ast }$.

\item[(H6)] $\sup \{\Phi _{0}(u):u\in X_{k}^{+}\oplus X^{\ast
}\oplus X^{-}\}=:M_{k}<\infty $ \ \ for every $k\geq 1$.

\item[(H7)] For each $k\geq 1$ there exist $n_{k}\geq 1$ and a
nondecreasing function $\ell _{k}:\mathbb{R}\to \mathbb{R}$ with the
following property: Given $n\geq n_{k}$, an $\iota $-equivariant map $\sigma
\in C^{0}(X_{k}^n,X^n)$ and an $R>0$ such that $\sigma (u)=u$ if $%
\| u\| >R,\ $there exist an $\iota $-equivariant map $\widetilde{
\sigma }\in C^{0}(X_{k+1}^n,X^n)$ and an $\widetilde{R}>R$ such that $%
\widetilde{\sigma }(u)=\sigma (u)$ if $u\in X_{k}^n$, $\widetilde{\sigma }
(u)=u$ if $\| u\| >\widetilde{R},\ $and$\ $
\begin{equation*}
\sup \Phi _{0}(\widetilde{\sigma }(X_{k+1}^n))\leq \ell _{k}(\sup \Phi
_{0}(\sigma (X_{k}^n))).
\end{equation*}
\end{itemize}

We shall prove the following statement.

\begin{theorem} \label{abs}
Assume that $\Phi $ satisfies (H1)-(H7). Then there
exists a sequence $(c_{k})$ of real numbers such that, if the sequence
\begin{equation}
\Big( \frac{c_{k+1}-c_{k}}{\mathrel{\mathop{\max }_{0\leq t\leq 1}}%
| \theta _{1}(t,c_{k+1})| +\mathrel{\mathop{\max }_{0\leq
t\leq 1}}| \theta _{2}(t,c_{k})| +1}\Big)  \label{seq}
\end{equation}
is unbounded, then $\Phi _{1}$ has an unbounded sequence of critical values.
\end{theorem}

The numbers $c_{k}$ are defined as follows (see (\ref{ck}) below): For each $%
n\in \mathbb{N}$, let
\begin{equation*}
c_{k}^n:=\inf_{\sigma \in \Gamma _{k}^n}\hbox{ }\sup_{u\in
X_{k}^n}\Phi _{0}(\sigma (u))
\end{equation*}
where $\Gamma _{k}^n$ is the set of maps $\sigma \in
C^{0}(X_{k}^n,X^n) $ with the following three properties:

\begin{itemize}
\item[$(i)$] $\sigma $ is $\iota $-equivariant, that is, $\sigma (\iota
(u))=\iota (\sigma (u))$ $\ $\ for each $u\in X_{k}^n$,

\item[$(ii)$] There exists $R>0$ such that $\ \sigma (u)=u$ $\ $if $\|
u\| >R$.

\item[$(iii)$] $\sigma (u)=u$ $\ $for each $u\in X_{n}^{\ast }$.
\end{itemize}

These values have the following linking property for $\Phi _{0}$.

\begin{lemma} \label{rigid}
Let $e\in X_{k+1}^{+}\setminus X_{k}^{+}$  and let
\begin{equation*}
\vartheta :\{v+se\in X_{k+1}^n:v\in X_{k}^n,\; s\in [
0,\infty )\}\to X^n
\end{equation*}
be a continuous map with the following properties:
\begin{itemize}
\item[(i)] $\vartheta \mid X_{k}^n$  is $\iota $-equivariant.

\item[(ii)]  There exists $R>0$ such that $\vartheta (u)=u$  if $\|u\| >R$.

\item[(iii)] $\vartheta (v)=v$ $\ $for all $v\in X_{n}^{\ast }$.
\end{itemize}
Then there exists  $(v_{0},s_{0})\in X_{k}^n\times [ 0,\infty )$
such that
\begin{equation*}
\Phi _{0}(\vartheta (v_{0}+s_{0}e))\geq c_{k+1}^n.
\end{equation*}
\end{lemma}

\begin{proof}
We extend $\vartheta $ to a map $\widetilde{\vartheta }:X_{k+1}^n
\to X^n$ as follows:
\begin{equation*}
\widetilde{\vartheta }(v+se):=\iota \vartheta (\iota v-se)\hbox{ \ \ \ if \ }
(v,s)\in X_{k}^n\times (-\infty ,0].
\end{equation*}
Since $\dim X_{k+1}^n=\dim X_{k}^n+1$ and since$\ \vartheta \mid
X_{k}^n$ \ is $\iota $-equivariant, $\widetilde{\vartheta }$\ is well
defined. By definition, $\widetilde{\vartheta }\in \Gamma _{k+1}^n$. Hence
there exists $u_{0}\in X_{k+1}^n$ with $\Phi _{0}(\widetilde{\vartheta }
(u_{0}))\geq c_{k+1}^n$. But $\Phi _{0}\circ \iota =\Phi _{0}$ by
assumption (H5). Therefore $\iota u_{0}$ also satisfies $\Phi _{0}(
\widetilde{\vartheta }(\iota u_{0}))\geq c_{k+1}^n$. So, without loss of
generality, $u_{0}=v_{0}+s_{0}e$ with $(v_{0},s_{0})\in X_{k}^n\times
[ 0,\infty )$.
\end{proof}

Since the inclusion $X_{k}^n\hookrightarrow X^n$ belongs to $\Gamma
_{k}^n$, assumption (H6) guarantees that
\begin{equation*}
c_{k}^n\leq \sup_{u\in X_{k}^n}\Phi _{0}(u)\leq M_{k}<\infty
\hbox{ \ \
\ for all }k,n\geq 1.
\end{equation*}
The numbers $c_{k}$ in Theorem \ref{abs} are defined as follows:
\begin{equation}
c_{k}:=\mathrel{\mathop{\lim \sup }_{n\to \infty }}c_{k}^n.
\label{ck}
\end{equation}
Note that $c_{k}\leq c_{k+1}$ for all $k\geq 1$.

We shall assume without loss of generality that  $c_{k}^n\to
c_{k} $ as $n\to \infty $. We also assume from now on that (H1)-(H7)
hold and that the sequence (\ref{seq}) is unbounded.
We wish to show that the linking property of $\Phi _{0}$ is preserved by the
flow of the path of functionals $\Phi _{t}$ in a suitable sense. We start
with the following easy observation.

\begin{lemma} \label{ro}
For every $b,\delta >0$ there exist $n_{0}\in \mathbb{N}$ and $\rho >0 $
such that
\begin{equation*}
\theta _{1}(t,\Phi _{t}(u))-\delta <\frac{\partial }{\partial t}\Phi
(u,t)<\theta _{2}(t,\Phi _{t}(u))+\delta
\end{equation*}
for every $(u,t)\in X^n\times [ 0,1]$, with $n\geq n_{0}$, $|
\Phi _{t}(u)| <b$ and $\| (\Phi _{t}^n)'(u)\|<\rho $.
\end{lemma}

The statement of the above lemma is an immediate consequence of (H1) and (H3).

Fix $\delta >0$. For $\theta _{1}$ and $\theta _{2}$ as in (H3) we consider
the flows $\zeta _{1},\zeta _{2}:[0,1]\times \mathbb{R}\to \mathbb{R}
$ defined by
\begin{gather*}
\zeta _{1}(0,s)=s \\
\frac{\partial }{\partial t}\zeta _{1}(t,s)=\theta _{1}(t,\zeta
_{1}(t,s))-\delta
\end{gather*}
and
\begin{gather*}
\zeta _{2}(0,s)=s \\
\frac{\partial }{\partial t}\zeta _{2}(t,s)=\theta _{2}(t,\zeta
_{2}(t,s))+\delta\,.
\end{gather*}
Following Bolle \cite{bo} we prove the following deformation lemma for the
path of functionals $\Phi _{t}$.

\begin{lemma} \label{defor}
For every pair of real numbers $d_{1}\leq d_{2}$ there exist $n_{0}\in \mathbb{N}$
and, for each $n\geq n_{0}$ and each $\nu =1,2$, there
exists a homotopy $\eta _{\nu }^n:X^n\times [ 0,1]\to X^n $ with the following
properties:
\begin{itemize}
\item[(i)] $\eta _{\nu }^n(u,0)=u$ for all $u\in X^n$.

\item[(ii)] $\eta _{\nu }^n(u,t)=u$ if either $\Phi _{t}(u)\leq
\,\mathrel{\mathop{\min }_{0\leq
t\leq 1}}\zeta _{\nu }(t,d_{1})-1$ or\\ $\Phi _{t}(u)\geq
\,\mathrel{\mathop{\max }_{
0\leq t\leq 1}}\zeta _{\nu }(t,d_{2})+1$.

\item[(iii)] $\eta _{\nu }^n(\cdot ,t):X^n\to X^n$ is a homeomorphism
for every $t\in [ 0,1]$.

\item[(iv)] If $c\in [ d_{1},d_{2}]$ and $\Phi _{0}(u)\geq c$, then $\Phi
_{t}(\eta _{1}^n(u,t))\geq \zeta _{1}(t,c)$ for all $t\in [ 0,1]$.

\item[(v)] If $c\in [ d_{1},d_{2}]$ and $\Phi _{0}(u)\leq c$, then $\Phi
_{t}(\eta _{2}^n(u,t))\leq \zeta _{2}(t,c)$ for all $t\in [ 0,1]$.
\end{itemize}
\end{lemma}

\begin{proof}
We extend Bolle's argument \cite{bo} to Banach spaces and $C^{1}$%
-functionals as follows. Let $M^n=\{(u,t)\in X^n\times [
0,1]:(\Phi _{t}^n)'(u)\neq 0\}$ and let $W^n:M^n\to
X^n$ be a pseudogradient vector field for the map $(u,t)\mapsto (\Phi
_{t}^n)'(u)$, that is, $W^n$ is locally Lipschitz continuous
and satisfies
\begin{equation}
\| W^n(u,t)\| \leq 2\| (\Phi _{t}^n)'(u)\| \quad\hbox{and}\quad
\left\langle (\Phi _{t}^n)'(u),W^n(u,t)\right\rangle
\geq \| (\Phi _{t}^n)'(u)\| ^2  \label{psgr}
\end{equation}
for every $(u,t)\in M^n$ \cite[Lemma 2.2]{w}. Set $\alpha _{\nu }=\min
\{\zeta _{\nu }(t,d_{1}):0\leq t\leq 1\}$ and $\beta _{\nu }=\max \{\zeta
_{\nu }(t,d_{2}):0\leq t\leq 1\}$. For $b=\max \{| \alpha _{\nu
}| ,| \beta _{\nu }| :\nu =1,2\}$ and $\delta $ as above we
choose $n_{0}\in \mathbb{N}$ and $\rho >0$ as in Lemma \ref{ro}. Let
$\lambda _{\nu },\mu \in C^{\infty }(\mathbb{R},[0,1])$ be such that $\lambda
_{\nu }\equiv 0$ on $(-\infty ,\alpha _{\nu }-\frac{1}{2}]\cup [ \beta
_{\nu }+\frac{1}{2},\infty )$ and $\lambda _{\nu }\equiv 1$ on $[\alpha
_{\nu },\beta _{\nu }]$, and$\ $ $\mu \equiv 0$ on
$[-\frac{\rho }{2},\frac{\rho }{ 2}]$ and $\mu \equiv 1$ on
$(-\infty ,-\rho ]\cup [ \rho ,\infty )$.

Fix $n\geq n_{0}$ and consider the vector fields $V_{\nu }^n:X^n\times
[ 0,1]\to X^n$ given by
\begin{align*}
&V_{1}^n(u,t)\\
&=4\Big( (\frac{\partial }{\partial t}\Phi )^{-}(u,t)+1+\theta
_{1}^{+}(t,\zeta _{1}(t,c))\Big) \lambda _{1}(\Phi _{t}(u))\mu (\|
W^n(u,t)\| )\frac{W^n(u,t)}{\| W^n(u,t)\| ^2}, \\
&V_{2}^n(u,t)\\
&=-4\Big( (\frac{\partial }{\partial t}\Phi )^{+}(u,t)+1+\theta
_{2}^{-}(t,\zeta _{2}(t,c))\Big) \lambda _{2}(\Phi _{t}(u))\mu
(\|W^n(u,t)\| )\frac{W^n(u,t)}{\| W^n(u,t)\| ^2},
\end{align*}
where $h^{\pm }:=\max \{\pm h,0\}\geq 0$. Note that $V_{\nu }^n(u,t)=0$ if
$\Phi _{t}(u)\notin [ \alpha _{\nu }-\frac{1}{2},\beta _{\nu }+\frac{1
}{2}]$ or $\| W^n(u,t)\| \leq \frac{\rho }{2}$. On the other
hand, if $\Phi _{t}(u)\in [ \alpha _{\nu }-\frac{1}{2},\beta _{\nu }+
\frac{1}{2}]$ and $\| W^n(u,t)\| \geq \frac{\rho }{2}$ then,
conditions (H2) and (\ref{psgr}) imply that
\begin{align*}
\| V_{\nu }^n(u,t)\| &\leq \frac{4(| (\frac{\partial }{
\partial t}\Phi)(u,t)| +1+| \theta _{\nu }(t,\zeta _{\nu
}(t,c))| )}{\| W^n(u,t)\| } \\
&\leq \frac{\widetilde{C}(\| (\Phi _{t}^n)'(u)\|
+1)(\| u\| +1)}{\| W^n(u,t)\| } \\
&\leq \widehat{C}(\| u\| +1)
\end{align*}
for some positive constants $\widetilde{C}$ and $\widehat{C}$. This, and the
fact that $V_{\nu }^n$ is locally Lipschitz continuous, imply the
existence of a global flow $\eta _{\nu }^n:X^n\times [0,1]\to X^n$ for
 $V_{\nu }^n$ given by
\begin{gather*}
\eta _{\nu }^n(u,0)=u \\
\frac{\partial }{\partial t}\eta _{\nu }^n(u,t)=V_{\nu }^n(\eta _{\nu
}^n(u,t),t)
\end{gather*}
Properties (i)-(iii) are immediate. We prove (iv): Let $u\in
X^n$ satisfy $\Phi _{0}(u)\geq c$ for some $c\in [ d_{1},d_{2}]$.
Set $f(t):=\Phi _{t}(\eta _{1}^n(u,t))$. Since $f(0)=\Phi _{0}(u)\geq
c=\zeta _{1}(0,c)$ it suffices to show that
\begin{equation}
f(t)=\zeta _{1}(t,c)\Longrightarrow f'(t)>\frac{\partial }{\partial
t}\zeta _{1}(t,c)=\theta _{1}(t,\zeta _{1}(t,c))-\delta .  \label{iv}
\end{equation}
So let us assume $f(t)=\zeta _{1}(t,c)$. Then $\lambda _{1}(f(t))=1$. Hence,
setting
\begin{equation*}
\varphi (t):=(\frac{\partial }{\partial t}\Phi )^{-}(\eta
_{1}^n(u,t),t)+1+\theta _{1}^{+}(t,\zeta _{1}(t,c)),
\end{equation*}
we obtain
\begin{align*}
f'(t)
&= \left\langle (\Phi _{t}^n)'(\eta _{1}^n(u,t)),V_{1}^n
 (\eta_{1}^n(u,t),t)\right\rangle +\frac{\partial }{\partial t}\Phi
 (\eta_{1}^n(u,t),t) \\
&= 4\varphi (t)\mu (\| W^n(\eta _{1}^n(u,t),t)\| )\frac{
\left\langle (\Phi _{t}^n)'(\eta _{1}^n(u,t)),W^n(\eta
_{1}^n(u,t),t)\right\rangle }{\| W^n(\eta _{1}^n(u,t),t)\|^2}\\
&\quad +\frac{\partial }{\partial t}\Phi (\eta _{1}^n(u,t),t) \\
&\geq \varphi (t)\mu (\| W^n(\eta _{1}^n(u,t),t)\| )+\frac{
\partial }{\partial t}\Phi (\eta _{1}^n(u,t),t)
\end{align*}
If $\| W^n(\eta _{1}^n(u,t),t)\| <\rho $ then
$\| (\Phi_{t}^n)'(\eta _{1}^n(u,t))\| <\rho $ and, by
Lemma \ref{ro},
\begin{equation*}
f'(t)\geq \frac{\partial }{\partial t}\Phi (\eta_{1}^n(u,t),t)
>\theta _{1}(t,\zeta _{1}(t,c))-\delta .
\end{equation*}
If $\| W^n(\eta _{1}^n(u,t),t)\| \geq \rho $ then
$\mu(\| W^n(\eta _{1}^n(u,t),t)\| )=1$, hence,
\begin{equation*}
f'(t)\geq \varphi (t)+\frac{\partial }{\partial t}
\Phi (\eta_{1}^n(u,t),t)\geq \theta _{1}(t,\zeta _{1}(t,c))
>\theta _{1}(t,\zeta_{1}(t,c))-\delta .
\end{equation*}
This proves (\ref{iv}). Therefore, $\eta _{1}^n$ satisfies (iv).
Similarly, $\eta _{2}^n$ satisfies (v).
\end{proof}

Given a subset $A$ of $X^n$ we define
\begin{align*}
\Gamma _{k}^n(A):=\big\{&\tau \in C^{0}(X^n,X^n):\tau (u)=u
\hbox{ if either }u\in A,\\
&\hbox{or }u\in X_{k+1}^n\hbox{ and }\| u\| \hbox{ is large}\}.
\end{align*}
Let $M\geq 0$ be as in (H5), and let $n_{k}\geq 1$ and $\ell _{k}:\mathbb{R}
\to \mathbb{R}$ be as in (H7). We now prove a linking property for $%
\Phi _{1}$.

\begin{lemma} \label{chh}
For every $k\in \mathbb{N}$ such that
\begin{equation*}
M+1\leq \zeta _{2}(t,c_{k})<\zeta _{1}(t,c_{k+1})\quad  \mbox{for all }t\in [ 0,1],
\end{equation*}
there exist $\varepsilon _{k}>0$, $m_{k}\geq n_{k}$ and, for each $n\geq
m_{k}$, two subsets $A_{k}^n\subset B_{k}^n$ of $X^n$ with the
following properties:
\begin{itemize}
\item[(a)] $\sup \Phi _{1}(A_{k}^n)\leq \zeta _{2}(1,c_{k}+\varepsilon
_{k})<\zeta _{1}(1,c_{k+1}-\varepsilon _{k})$.

\item[(b)] $\sup \Phi _{1}(B_{k}^n)\leq \zeta _{2}(1,\ell _{k}(c_{k}+\varepsilon
_{k}))$.

\item[(c)]  $\inf_{\tau \in \Gamma _{k}^n(A_{k}^n)}\sup \Phi _{1}(\tau
(B_{k}^n))\geq \zeta _{1}(1,c_{k+1}-\varepsilon _{k})$.
\end{itemize}
\end{lemma}

\begin{proof}
Fix $0<\varepsilon _{k}<1$ such that
\begin{equation}
\zeta _{2}(t,c_{k}+\varepsilon _{k})<\zeta _{1}(t,c_{k+1}-\varepsilon _{k})
\quad \hbox{for all }t\in [ 0,1].  \label{a}
\end{equation}
Let $m_{k}\geq n_{k}$ be such that, for each $n\geq m_{k}$,
\begin{equation*}
c_{k}^n<c_{k}+\varepsilon _{k}\quad\hbox{and}\quad c_{k+1}^n>c_{k+1}-
\varepsilon _{k},
\end{equation*}
and there exist homotopies $\eta _{\nu }^n:X^n\times [0,1]\to X^n$ which satisfy
(i)-(v) of Lemma \ref{defor} for
$d_{1}=d_{2}=c_{k+1}-\varepsilon _{k}$ if $\nu =1$, and for$\
d_{1}=c_{k}+\varepsilon _{k}$, $d_{2}=\ell _{k}(c_{k}+\varepsilon _{k})$ if $%
\nu =2$. In particular,
\begin{gather}
\Phi _{t}(\eta _{1}^n(u,t)) \geq \zeta _{1}(t,c_{k+1}-\varepsilon _{k})\quad
\hbox{if }\Phi _{0}(u)\geq c_{k+1}-\varepsilon _{k}  \label{b} \\
\Phi _{t}(\eta _{2}^n(u,t)) \leq \zeta _{2}(t,c)\quad\hbox{if }
\Phi_{0}(u)\leq c\quad\hbox{and}\quad c_{k}+\varepsilon _{k}
\leq c\leq \ell_{k}(c_{k}+\varepsilon _{k})  \label{b'}
\end{gather}
for all $t\in [ 0,1] $.  Fix $n\geq m_{k}$ and choose $\sigma\in \Gamma _{k}^n$
such that
\begin{equation*}
\sup \Phi _{0}(\sigma (X_{k}^n))\leq c_{k}+\varepsilon _{k}.
\end{equation*}
By assumption (H7), $\sigma $ has an extension
$\widetilde{\sigma }\in \Gamma _{k+1}^n$ such that
\begin{equation*}
\sup \Phi _{0}(\widetilde{\sigma }(X_{k+1}^n))\leq \ell
_{k}(c_{k}+\varepsilon _{k}).
\end{equation*}
Inequalities (\ref{a}), (\ref{b}) and (\ref{b'}) imply that
\begin{equation}
(\eta _{2}^n)_{t}(\sigma (X_{k}^n))\cap (\eta _{1}^n)_{t}(\Phi
_{0}^n)^{>c_{k+1}-\varepsilon _{k}}=\emptyset\quad \hbox{for all }t\in
[ 0,1],  \label{d}
\end{equation}
where  $(\Phi _{0}^n)^{>c_{k+1}-\varepsilon _{k}}:=\{u\in X^n:\Phi
_{0}(u)>c_{k+1}-\varepsilon _{k}\}$. Choose $e\in X_{k+1}^{+}\setminus
X_{k}^{+}$ and define
\begin{gather*}
A_{k}^n :=\{(\eta _{2}^n)_{1}(\sigma (u)):u\in X_{k}^n\} \\
B_{k}^n :=\{(\eta _{2}^n)_{1}(\widetilde{\sigma }(u+te)):u\in
X_{k}^n,\; t\geq 0\}.
\end{gather*}
It follows from (\ref{b'}) that
\begin{equation*}
\sup \Phi _{1}(A_{k}^n)\leq \zeta _{2}(1,c_{k}+\varepsilon _{k})
\quad\hbox{and}\quad
\sup \Phi _{1}(B_{k}^n)\leq \zeta _{2}(1,\ell_{k}(c_{k}+\varepsilon _{k})).
\end{equation*}
Thus (a) and (b) hold. Let us prove (c). For every
$\tau \in \Gamma _{k}^n(A_{k}^n)$, the function
$\vartheta :\{u+te:u\in X_{k}^n,\hbox{ }t\in [ 0,\infty )\}\to X^n$,
defined by
\begin{equation*}
\vartheta (u+te)=\begin{cases}
(\eta _{1}^n)_{2t}^{-1}\circ (\eta _{2}^n)_{2t}\circ \sigma (u) &
\hbox{if }0\leq t\leq 1/2 \\
(\eta _{1}^n)_{1}^{-1}\circ \tau \circ (\eta _{2}^n)_{1}\circ \widetilde{
\sigma }(u+(2t-1)e) & \hbox{if }1/2\leq t\leq 1
\end{cases}
\end{equation*}
satisfies the hypotheses of Lemma \ref{rigid}. Hence there exists \ $%
(u_{0},t_{0})\in X_{k}^n\times [ 0,\infty )$ \ with
\begin{equation}
\Phi _{0}(\vartheta (u_{0}+t_{0}e))\geq c_{k+1}^n>c_{k+1}-\varepsilon _{k}.
\label{link}
\end{equation}
If $t_{0}\leq 1/2$ then $(\eta _{1}^n)_{2t_{0}}(\vartheta
(u_{0}+t_{0}e))=(\eta _{2}^n)_{2t_{0}}(\sigma (u_{0}))$, contradicting
(\ref{d}). Therefore, $t_{0}>1/2$ and $(\eta _{1}^n)_{1}(\vartheta
(u_{0}+t_{0}e))=\tau \left[ (\eta _{2}^n)_{1}(\widetilde{\sigma }
(u_{0}+(2t_{0}-1)e)\right] $. Inequalities (\ref{b}) y (\ref{link}) yield
\begin{equation*}
\Phi _{1}(\tau \left[ (\eta _{2}^n)_{1}(\widetilde{\sigma }
(u_{0}+(2t_{0}-1)e)\right] )=\Phi _{1}((\eta _{1}^n)_{1}(\vartheta
(u_{0}+t_{0}e)))\geq \zeta _{1}(1,c_{k+1}-\varepsilon _{k})
\end{equation*}
and, since $(\eta _{2}^n)_{1}(\widetilde{\sigma }(u_{0}+(2t_{0}-1)e)\in
B_{k}^n$, it follows that
\begin{equation*}
\sup \Phi _{1}(\tau (B_{k}^n))\geq \zeta _{1}(1,c_{k+1}-\varepsilon _{k}).
\end{equation*}
This proves (c).
\end{proof}

We now show that, if the sequence (\ref{seq}) is unbounded, the hypothesis
of Lemma \ref{chh} holds for infinitely many $k$'s.

\begin{lemma} \label{unb}
If the sequence (\ref{seq}) is unbounded, then the sequences
\begin{equation*}
\Big( \mathrel{\mathop{\min }_{0\leq t\leq 1}}(\zeta
_{1}(t,c_{k+1})-\zeta _{2}(t,c_{k}))\Big) \quad\mbox{and}\quad
\Big(\mathrel{\mathop{\min }_{0\leq t\leq 1}}\zeta _{2}(t,c_{k})\Big)
\end{equation*}
are unbounded above.
\end{lemma}

\begin{proof}
The flows $\zeta _{\nu }$ satisfy $| s-\zeta _{\nu }(t,s)| \leq
a(\max \{| \theta _{\nu }(t,s)| :t\in [ 0,1]\}+\delta )$
for some constant $a>0$ (cf. for example \cite{sch}). Therefore,
\begin{equation*}
0\leq c_{k+1}-c_{k}\leq \zeta _{1}(t,c_{k+1})-\zeta _{2}(t,c_{k})+a( %
\mathrel{\mathop{\max }_{0\leq t\leq 1}}| \theta
_{1}(t,c_{k+1})| + \max _{0\leq t\leq 1}
| \theta _{2}(t,c_{k})| +2\delta ).
\end{equation*}
Since the sequence (\ref{seq}) is unbounded, our claim follows.
\end{proof}

\begin{proof}[Proof of Theorem \ref{abs}]
By Lemma \ref{unb}, passing to a
subsequence if necessary, we may assume that
\begin{equation*}
M+1<\zeta _{2}(t,c_{k})<\zeta _{1}(t,c_{k+1})\hbox{ \ \ for all \ }t\in
[ 0,1],\hbox{ }k\geq 1.
\end{equation*}
Let $A_{k}^n\subset B_{k}^n\subset X^n$, $n\geq m_{k}$, be as in
Proposition \ref{chh}, and let
\begin{equation*}
\widetilde{c}_{k}^n:=\inf_{\tau \in \Gamma _{k}^n(A_{k}^n)}\hbox{ }
\sup_{u\in B_{k}^n}\Phi _{1}(\tau (u))
\end{equation*}
Then
\begin{equation}
\zeta _{2}(1,c_{k})<\zeta _{1}(1,c_{k+1})\leq \widetilde{c}_{k}^n
\leq \zeta _{2}(1,\ell _{k}(c_{k}+1))\quad\hbox{for all  }n\geq m_{k}.
\label{remf}
\end{equation}
Define
\begin{equation*}
\widetilde{c}_{k}:=\limsup_{n\to \infty} \widetilde{c}_{k}^n.
\end{equation*}
It follows from assumption (H1) and Proposition 2.6 (b) in \cite{bc} that $%
\widetilde{c}_{k}$ is a critical value of $\Phi _{1}:X\to X$. Since $%
\widetilde{c}_{k}\geq \zeta _{2}(1,c_{k})$, Lemma \ref{unb} implies that the
sequence $(\widetilde{c}_{k})$ is unbounded.
\end{proof}

We conclude this section with some remarks.

\subsection*{Remarks}
(a)\; If $\dim (X^{\ast }\oplus
X^{-})<\infty $ then assumptions (H6) and (H7) follow easily from assumption
(H4).

\noindent (b)\; If $X^{\ast }=\{0\}$, assumptions (H2)-(H5) are the same as
those of Theorem 2.2 in \cite{bgt}, and our assumption (H1) is stronger than
the one given there. But, if $\dim (X^{-})=\infty $, the minimax levels $%
c_{k}$ as defined by Bolle, Ghoussoub and Tehrani in \cite{bgt} will all be
zero. So their result does not yield critical values in the strongly
indefinite situation.

\noindent (c)\; Our definition of $c_{k}$ allows us to take advantage of
the topology of the sublevel sets of the semidefinite approximations $\Phi
_{0}^n$ of $\Phi _{0}$ and, in particular, to apply Morse theory methods
(as was done by Bahri-Lions \cite{bl} and Tanaka \cite{t}) to estimate the
growth of the $c_{k}$'s and derive conditions for the unboundedness of ( \ref%
{seq}), see Lemma \ref{lb} below.

\noindent (d)\; Assumption (H1) is the obvious extension to paths of
functionals of the (PS)$^{\ast }$-condition introduced by Bahri and
Berestycki \cite{bb1} and Li and Liu \cite{ll}. It yields critical values $%
\widetilde{c}_{k}$ of $\Phi _{1}$ provided that the minimax values $%
\widetilde{c}_{k}^n$ of its approximations $\Phi _{1}^n$ are uniformly
bounded. This is where (H7) comes into play.

\noindent (e)\; Assumption (H7) requires some knowledge on the topology of
the sublevel sets of the approximations of $\Phi _{0}$. Note that every $%
\iota $-equivariant map $\sigma \in C^{0}(X_{k}^n,X^n)$ such that $%
\sigma (u)=u$ for $\| u\| $ large enough,$\ $has an $\iota $
-equivariant extension $\widetilde{\sigma }\in C^{0}(X_{k+1}^n,X^n)$
such that $\widetilde{\sigma }(u)=u$ for $\| u\| $ sufficiently
large. The key point in assumption (H7) is that
\begin{equation*}
\sup \Phi _{0}(\widetilde{\sigma }(X_{k+1}^n))\leq \ell _{k}(\sup \Phi
_{0}(\sigma (X_{k}^n)))
\end{equation*}
for some function $\ell _{k}$ which does not depend on $n$. In fact, in our
application the function $\ell _{k}$ will be linear and will be also
independent of $k$, see Proposition \ref{H7} below.

\noindent (f)\; It follows from (\ref{remf}) that the critical values
$\widetilde{c}_{k}$ of the perturbed functional $\Phi _{1}$ satisfy
\begin{equation*}
\zeta _{2}(1,c_{k})<\zeta _{1}(1,c_{k+1})\leq \widetilde{c}_{k}
\leq \zeta_{2}(1,\ell _{k}(c_{k}+1)).
\end{equation*}
This inequality will be used to obtain estimates on the energy of the
solutions of problem $(\wp )$, see Theorem \ref{homo}.

\section{Strongly indefinite elliptic systems}

We apply Theorem \ref{abs} to obtain infinitely many solutions of the
elliptic system \eqref{P}. The variational setting is as follows:
Let $0<\lambda _{1}\leq \dots \leq \lambda _{j}\leq \dots $ be the
Dirichlet eigenvalues of $-\Delta $ on $H_{0}^{1}(\Omega )$
counted with their multiplicity, and let $e_{j}\in H_{0}^{1}(\Omega )$ be
the eigenfunction which corresponds to $\lambda _{j}$ with $|
e_{j}| _{2}=1$. Consider the Banach space $H_{0}^{1}(\Omega )\cap
L^{q}(\Omega )$ equipped with the norm
$\| v\| _{(q)}:=(| \nabla v| _{2}^2+|v| _{q}^2)^{1/2}$, where
$| \cdot | _{r}$ denotes the usual norm in $L^{r}$.
Let $V^{q}(\Omega )$ be the closure of $%
span\{e_{n}:n\geq 1\}$ in $H_{0}^{1}(\Omega )\cap L^{q}(\Omega )$\ with
respect to the $\| v\| _{(q)}$-norm.
Then $V^{q}(\Omega )$ is a Banach space. Since the eigenfunctions satisfy
$|e_{n}|_{\infty }^2\leq C\lambda _{n}^{N/2}$ for some positive constant $C$
\cite[Chap. IV, Theorem 8]{chavel}, integrating by parts one can easily show
that the Fourier coefficients of a function $\varphi \in C_{0}^{2m}(\Omega )$
decrease as $\lambda _{n}^{-m}$. It follows that
$C_{0}^{\infty }(\Omega )\subset V^{q}(\Omega )$.

Let $X$ be the direct sum
\begin{equation*}
X:=H_{0}^{1}(\Omega )\oplus V^{q}(\Omega )
\end{equation*}
We denote the elements of $X$ by $z=(u,v)$, and their norm by
\begin{equation*}
\| z\| _{q}:=(| \nabla u| _{2}^2+\| v\|_{(q)}^2)^{1/2}
=(| \nabla u| _{2}^2+| \nabla v| _{2}^2+| v| _{q}^2)^{1/2}.
\end{equation*}
Set
\begin{gather*}
X_{k}^{+} :=\mathop{\rm span}\{e_{1},...,e_{k}\}\subset H_{0}^{1}(\Omega )=:X^{+}, \\
X_{n}^{-} :=\mathop{\rm span}\{e_{1},...,e_{n}\}\subset V^{q}(\Omega )=:X^{-},
\end{gather*}
and $X^{\ast }:=\{0\}$. The orthogonal projection $H_{0}^{1}(\Omega
)\to \mathop{\rm span}\{e_{1},\dots,e_{n}\}\subset H_{0}^{1}(\Omega )$
restricts to a continuous operator
\begin{equation*}
P_{n}:V^{q}(\Omega )\to X_{n}^{-}
\end{equation*}%
which satisfies $P_{n}v\to v$ in $V^{q}(\Omega )$ as
$n\to \infty $ for every $v\in V^{q}(\Omega )$ because, by definition,
$\cup _{n\geq 1}X_{n}^{-}$ is dense in $X^{-}$.

Set
\begin{gather*}
H(x,z,t):=\frac{1}{p}| u| ^p+\frac{1}{q}| v|
^{q}+tf(x,z),\quad x\in \Omega ,\hbox{ }z=(u,v)\in \mathbb{R}^2, \\
\Phi (z,t):=\frac{1}{2}\int_{\Omega }\left( | \nabla u|
^2-| \nabla v| ^2\right) dx-\int_{\Omega }H(x,z,t)dx.
\end{gather*}
Theorem \ref{intro}\ is a special case of the following result.

\begin{theorem} \label{homo}
Let $p\in (2,2^{\ast })$, $q\in (2,\infty )$, and $f\in C^{1}(
\overline{\Omega }\times \mathbb{R}^2,\mathbb{R})$. If there exist $d>0$, $%
\gamma \in [ 0,1)$, $\gamma <\min \{ ( \frac{1}{p}-\frac{1}{
2^{\ast }}) N, \frac{q}{q-1}( \frac{2^{\ast }-1}{2^{\ast }})\} $ such that
\begin{gather*}
| f(x,z)| +| f_{z}(x,z)z| \leq d(| u|
^{\gamma p}+| v| ^{\gamma q}+1), \\
| f_{z}(x,z)| \leq d(| u| ^{\gamma (p-1)}+|
v| ^{\gamma (q-1)}+1)
\end{gather*}
for all $x\in \Omega $, $z=(u,v)\in \mathbb{R}^2$, then problem \eqref{P}
 has a sequence of solutions $z_{k}=(u_{k},v_{k})$ which satisfy
\begin{equation*}
C_{1}k^{\nu }\leq \Phi _{1}(z_{k})\leq C_{2}k^{\nu }\hbox{,}
\end{equation*}
with $\nu :=\frac{2p}{N(p-2)}$, $C_{1}, C_{2}>0$.
\end{theorem}

The assumptions of Theorem \ref{homo} guarantee that the functional $\Phi $\
is well defined and of class $C^{1}$. The critical points of $\Phi
_{1}:=\Phi (\cdot,1)$ are weak solutions of \eqref{P}. As in section 2
we set
\begin{equation*}
X^n:=X^{+}\oplus X_{n}^{-},\quad X_{k}^n:=X_{k}^{+}\oplus X_{n}^{-},
\end{equation*}
and write $\Phi _{t}^n:X^n\to \mathbb{R}$ for the functional
$\Phi _{t}^n(z)=\Phi (z,t)$, $z\in X^n$. We now show that $\Phi $
satisfies assumptions (H1)-(H7) of Theorem \ref{abs}. In order to prove (H1)
we need the following lemma.

\begin{lemma} \label{bdd}
Let $z_{k}\in X^{n_{k}}$, $t_{k}\in [ 0,1]$ be such that $n_{k}\to \infty $,
$\Phi _{t_{k}}(z_{k})\to c$ and $(\Phi _{t_{k}}^{n_{k}})'(z_{k})\to 0$ as
$k\to \infty $. Then $(z_{k})$ is bounded in $X$.
\end{lemma}

\begin{proof}
Our assumptions on $f$ yield
\begin{equation}
\frac{1}{2}H_{z}(x,z,t)z-H(x,z,t)\geq a_{1}\left( | u|^p+| v| ^{q}\right) -a_{2}.
 \label{main}
\end{equation}
Therefore, for $k$ large enough,
\begin{equation}
| u_{k}| _{p}^p+| v_{k}|_{q}^{q}\leq a_{3}(\Phi _{t_{k}}(z_{k})
-\frac{1}{2}(\Phi _{t_{k}}^{n_{k}})'(z_{k})z_{k}+1)\leq a_{4}(1+\|z_{k}\| _{q}).  \label{eq1}
\end{equation}
By assumption, $\eta =q/(q-\gamma (q-1))<2^{\ast }$. So, using H\"{o}lder's
inequality, Sobolev's embedding theorem and inequality (\ref{eq1}), we
obtain
\begin{equation}
\int_{\Omega }| v_{k}| ^{\gamma (q-1)}| u_{k}|
\leq | v_{k}| _{q}^{\gamma (q-1)}| u_{k}| _{\eta }\leq a_{5}(1
+\Vert z_{k} \Vert _{q}^{1+[\gamma (q-1)/q]}).  \label{eqII}
\end{equation}
Our assumptions on $f$ and $\gamma $, together with (\ref{eq1}) and (\ref{eqII}),
 imply
\begin{align*}
\int_{\Omega }H_{u}(x,z_{k},t_{k})u_{k}
&=| u_{k}|_{p}^p+t_{k}\int_{\Omega }f_{u}(x,z_{k})u_{k} \\
&\leq | u_{k}| _{p}^p+d\int_{\Omega }(|u_{k}| ^{\gamma (p-1)}+| v_{k}| ^{\gamma
(q-1)}+1)| u_{k}| \\
&\leq a_{6}(1+\| z_{k}\| _{q}^{\sigma })
\end{align*}
with $1<\sigma <2$. They also imply
\begin{align*}
&-\int_{\Omega }H_{v}(x,z_{k},t_{k})v_{k}\\
&=\int_{\Omega}H_{u}(x,z_{k},t_{k})u_{k}-\int_{\Omega }H_{z}(x,z_{k},t_{k})z_{k} \\
&\leq a_{6}(1+\| z_{k}\| _{q}^{\sigma })-\int_{\Omega
}(| u_{k}| ^p+| v_{k}| ^{q})
+d\int_{\Omega }(| u_{k}| ^{\gamma p}+|v_{k}| ^{\gamma q}+1) \\
&\leq a_{6}(1+\| z_{k}\| _{q}^{\sigma })-c_{2}(|
u_{k}| _{p}^p+| v_{k}| _{q}^{q})+c_{3} \\
&\leq a_{7}(1+\| z_{k}\| _{q}^{\sigma }).
\end{align*}
We conclude that, for $k$ large enough,
\begin{gather}
| \nabla u_{k}| _{2}^2 =(\Phi_{t_{k}}^{n_{k}})'(z_{k})(u_{k},0)+\int_{\Omega
}H_{u}(x,z_{k},t_{k})u_{k}\leq a_{7}(1+\| z_{k}\|
_{q}^{\sigma }),  \label{eq2} \\
| \nabla v_{k}| _{2}^2 =-(\Phi_{t_{k}}^{n_{k}})'(z_{k})(0,v_{k})-\int_{\Omega
}H_{v}(x,z_{k},t_{k})v_{k}\leq a_{7}(1+\| z_{k}\|
_{q}^{\sigma }).  \label{eq3}
\end{gather}
Inequalities (\ref{eq1}), (\ref{eq2}) and (\ref{eq3}) yield
\begin{equation*}
\Vert z_{k}\Vert _{q}^2\leq a_{8}(1+\Vert z_{k}\Vert_{q}^{\sigma })
\end{equation*}
with $\sigma <2$. Therefore $(z_{k})$ must be bounded in $X$.
\end{proof}

\begin{proposition} \label{h1}
The function $\Phi $ satisfies (H1).
\end{proposition}

\begin{proof}
Let $z_{k}\in X^{n_{k}}$, $t_{k}\in [ 0,1]$ be such that
$n_{k}\to \infty $, $t_{k}\to t$, $\Phi
_{t_{k}}(z_{k})\to c$ and $(\Phi _{t_{k}}^{n_{k}})'(z_{k})\to 0$
 as $k\to \infty $. Write $z_{k}=(u_{k},v_{k}). $ Since $(z_{k})$
 is bounded in $X$, $u_{k}\rightharpoonup u$ weakly in $H_{0}^1(\Omega)$
 and $v_{k}\rightharpoonup v$ weakly in $H_{0}^1(\Omega)\cap L^q(\Omega)$.
 Hence $u_{k}\to u$ strongly in $L^{s}(\Omega )$ for every
$s\in [1,2^{\ast })$ and, by interpolation, $v_{k}\to v$ strongly in
$L^{s}(\Omega )$ for every $s\in [ 1,\max \{2^{\ast },q\})$. Our
assumptions and H\"{o}lder's inequality yield
\begin{equation*}
\big| \int_{\Omega }H_{u}(x,z_{k},t_{k})(u_{k}-u)\big|
\leq  a_{1}\big( | u_{k}| _{p}^{p-1}| u_{k}-u|
_{p}+| v_{k}| _{q}^{\gamma (q-1)}| u_{k}-u| _{\eta
}+| u_{k}-u| _{1}\big) ,
\end{equation*}
with $\eta =q/(q-\gamma (q-1))<2^{\ast }$. Hence,
\[
\int_{\Omega }\nabla u_{k}\nabla (u_{k}-u)
=(\Phi _{t_{k}}^{n_{k}})'(z_{k})(u_{k}-u,0)+\int_{\Omega
}H_{u}(x,z_{k},t_{k})(u_{k}-u)
\longrightarrow 0
\]
as $k\to \infty$; therefore, $u_{k}\to u$ strongly in $H_{0}^{1}(\Omega )$.

The projection $P_{n}:V^{q}(\Omega )\to V^{q}(\Omega )_{n}$
satisfies $P_{n}v\to v$ in $V^{q}(\Omega )$ as $n\to \infty $
for every $v\in V^{q}(\Omega )$. Thus, $(v_{k}-P_{n_{k}}v)\to 0$ in $%
L^{s}(\Omega )$ for every $s\in [ 1,\max \{2^{\ast },q\})$. As above
we have
\begin{align*}
&\big| \int_{\Omega }t_{k}f_{v}(x,z_{k})(v_{k}-P_{n_{k}}v)| \\
&\leq a_{1}\big( | u_{k}| _{p}^{p-1}|
v_{k}-P_{n_{k}}v| _{p}+| v_{k}| _{q}^{\gamma (q-1)}|
v_{k}-P_{n_{k}}v| _{\eta }+| v_{k}-P_{n_{k}}v| _{1}\big) ,
\end{align*}
with $\eta =q/(q-\gamma (q-1))<2^{\ast }$. Hence,
\begin{equation*}
\int_{\Omega }t_{k}f_{v}(x,z_{k})(v_{k}-P_{n_{k}}v)\to 0
\quad\hbox{as }k\to \infty .
\end{equation*}
Therefore,
\begin{align*}
| \nabla v_{k}| _{2}^2-| \nabla v|_{2}^2+o(1)
&=\int_{\Omega }\nabla v_{k}\nabla (v_{k}-P_{n_{k}}v) \\
&=-(\Phi _{t_{k}}^{n_{k}})'(z_{k})(0,v_{k}-P_{n_{k}}v)-\int_{\Omega
 }H_{v}(x,z_{k},t_{k})(v_{k}-P_{n_{k}}v) \\
& =  o(1)-\int_{\Omega }| v_{k}| ^{q-2}v_{k}(v_{k}-P_{n_{k}}v) \\
& =  o(1)+\int_{\Omega }| v_{k}| ^{q-2}v_{k}(v-v_{k}) \\
& \leq  o(1)+| v| _{q}^{q}-| v_{k}| _{q}^{q},
\end{align*}
where $o(1)\to 0$ as $k\to \infty $. It follows that
\begin{equation*}
0\leq \liminf | \nabla v_{k}| _{2}^2-| \nabla v|
_{2}^2\leq \limsup | \nabla v_{k}| _{2}^2-| \nabla
v| _{2}^2\leq | v| _{q}^{q}-\liminf | v_{k}|_{q}^{q}\leq 0.
\end{equation*}
Hence, up to a subsequence, $v_{k}\to v$ strongly in $V^{q}(\Omega)$.
 This proves that $z_{k}\to z$ strongly in $X$.
In particular, $\Phi _{t_{k}}'(z_{k})\zeta \to \Phi_{t}'(z)\zeta $ for
every $\zeta \in X$. Hence, $\Phi _{t}'(z)\zeta =0$ for every
$\zeta \in \cup _{n\geq 1}X^n$. Since $\cup_{n\geq 1}X^n$ is dense in $X$, $z$
is a critical point of $\Phi _{t}$.
\end{proof}

\begin{proposition} \label{h3}
The function $\Phi $ satisfies (H2)-(H6) with $\theta
_{2}(t,s)=A(s^2+1)^{\gamma /2}=-\theta _{1}(t,s)$,  $A>0$.
\end{proposition}

\begin{proof}
Our assumptions on $f$ yield
\begin{equation}
| \frac{\partial }{\partial t}\Phi (z,t)| \leq \int_{\Omega
}| f(x,z)| \leq d_{1}\big| \Phi _{t}(z)-\frac{1}{2}\Phi
_{t}'(z)z+1\big| ^{\gamma }.  \label{part}
\end{equation}
If $z\in X^n$ and $| \left( \Phi _{t}^n\right)(z)| \leq b$, this
inequality implies that
\begin{equation*}
| \frac{\partial }{\partial t}\Phi (z,t)| \leq d_{2}(\|
\left(\Phi _{t}^n\right)'(z)\| _{(X^n)'}\|
z\| _{q}+1).
\end{equation*}
This proves (H2). If $\Phi _{t}'(z)=0$ then (\ref{part}) yields
\begin{equation*}
| \frac{\partial }{\partial t}\Phi (z,t)| \leq d_{1}| \Phi
_{t}(z)+1| ^{\gamma }\leq A(\Phi _{t}(z)^2+1)^{\gamma /2}.
\end{equation*}
This proves (H3) with $A(\Phi _{t}(z)^2+1)^{\gamma /2}=\theta _{2}(t,\Phi
_{t}(z))=-\theta _{1}(t,\Phi _{t}(z))$. \ Properties (H4)-(H6) are also easy.
\end{proof}

We now show that (H7) holds with $\ell (t)=\alpha t+\beta $, $\alpha ,\beta $
positive constants independent of $k$. We split the proof into several
lemmas.The first two were proved in \cite{cc}. We sketch their proofs for
the readers convenience.

\begin{lemma}\label{lem}
Let $\Omega $ be a bounded smooth domain in $\mathbb{R}^{N}$.
Then there exists $a\in \mathbb{R}$ with the following properties:
\begin{itemize}
\item[(i)] \ $(x',a)\in \Omega $ for some $x'\in \mathbb{R}%
^{N-1}. $

\item[(ii)] If $(x',b)\in \Omega $ and $b\geq a$, then $(x',t)\in
\Omega $ for all $a\leq t\leq b$.
\end{itemize}
\end{lemma}

\begin{proof}
Let $e_{N}=(0,\dots ,0,1)\in \mathbb{R}^{N}$, let $M=\max \{x\cdot e_{N}:x\in
\overline{\Omega }\}$ and let $K=\{x\in \overline{\Omega }:x\cdot e_{N}=M\}$.
Let $\nu :\partial \Omega \to \mathbb{R}^{N}$ be the outer unit
normal field, and let $\mathcal{O}=\{x\in \partial \Omega :\nu (x)\cdot
e_{N}>0\}$. Then $\mathcal{O}$ is an open neighborhood of $K$ in $\partial
\Omega $ and, since $K$ is compact, there is an $a<M$ such that the set $%
A=\{x\in \partial \Omega :x\cdot e_{N}\geq a\}\subset \mathcal{O}$. Thus,
for every $(x',t)\in A$ there exists $\varepsilon >0$ such that $%
(x',s)\notin \Omega $ if $t<s<t+\varepsilon $ and $(x',s)\in \Omega $
if $t-\varepsilon <s<t$. It follows that, for every $(x',t)\in A$,
\begin{equation*}
(\{x'\}\times [ a,M])\cap \partial \Omega =\{(x',t)\} \quad
\hbox{and}\quad \{x'\}\times [ a,t)\subset \Omega
\end{equation*}
as claimed.
\end{proof}

Set
\begin{gather*}
I(u) :=\frac{1}{2}\int_{\Omega }| \nabla u| ^2-\frac{1}{p} \int_{\Omega }| u| ^p,\\
J(v):=\frac{1}{2} \int_{\Omega }| \nabla v| ^2+\frac{1}{q}\int_{\Omega }|v| ^{q}, \\
I^{\#}(u) :=2\int_{\Omega }| \nabla u| ^2-\frac{1}{p} \int_{\Omega }| u| ^p.
\end{gather*}

\begin{lemma} \label{rep}
There is an even continuous function $\tau :H_{0}^{1}(\Omega
)\to [ 0,\infty )$ with the following properties:
\begin{itemize}
\item[(i)] $I([(1-s)+s\tau (u)]u)\leq I(u)$ \ for every $u\in H_{0}^{1}(\Omega)$,
 $0\leq s\leq 1$.

\item[(ii)]  If $\ I^{\#}(u)\leq 0$ $\ $then $\ \tau (u)=1$.

\item[(iii)] If $\ 2I(u)\leq \max_{t\geq 0}I(tu)$ \ then $I^{\#}(\tau (u)u)\leq 0$.

\item[(iv)] $\ I^{\#}(\tau (u)u)\leq \max \{\alpha I(u),0\}$  with $\alpha
:=2^{(3p-2)/(p-2)}$.
\end{itemize}
\end{lemma}

\begin{proof}
Fix $v\in H_{0}^{1}(\Omega )$ with $\| v\| =1$ and define \ $%
0<t_{v}^{-}<\widehat{t}_{v}<t_{v}^{+}<T_{v}<\infty $ \ as follows:
\begin{gather*}
I(\widehat{t}_{v}v) =\max_{t\geq 0}I(tv), \\
2I(tv) \geq \max_{t\geq 0}I(tu)\Longleftrightarrow t\in [t_{v}^{-},t_{v}^{+}], \\
2(T_{v})^2 =\frac{1}{p}| v| _{p}^p(T_{v})^p.
\end{gather*}
For $t\geq 0$ let $\rho (tv)$ be the piecewise linear function such that $%
\rho (tv)=0$ if \ $0\leq t\leq t_{v}^{-}$, $\rho (\widehat{t}_{v}v)=\widehat{
t}_{v}$, $\rho (tv)=T_{v}$ if $t_{v}^{+}\leq t\leq T_{v}$, and $\rho (tv)=t$
if $t\geq T_{v}$. For $u=tv\in H_{0}^{1}(\Omega )$ with $\| v\|
=1$, $t\geq 0$, define
\begin{equation*}
\tau (u)=\frac{\rho (tv)}{t}.
\end{equation*}
This function is continuous and satisfies (i), (ii), (iii), (iv).
\end{proof}

\begin{lemma} \label{mainlem}
There exist $\alpha ,\beta >0$, depending only on $\Omega $
and $p$, with the following property: For every map
$\varphi :X_{k}^n\to X^n$ such that $\varphi (z)=z$ if
$\| z\| \geq R$, there exist a map
$\psi :X_{k}^n\times [ 0,\infty )\to X^n$ and an $R'\geq R$ which satisfy:
\begin{itemize}
\item[(i)]  $\psi (z,0)=\varphi (z)$  for every $z\in X_{k}^n$.

\item[(ii)]  $\Phi _{0}(\psi (z,s))\leq -1$  if $\| z\| \geq R'$  or  $s\geq R'$.

\item[(iii)] $\Phi _{0}(\psi (z,s))\leq \max \{\alpha \Phi _{0}(\varphi (z))+\beta
,0\}$  for every $z\in X_{k}^n$, $s\geq 0$.
\end{itemize}
\end{lemma}

\begin{proof}
Let $a$ be as in Lemma \ref{lem}. We may assume without loss of generality
that $a=0$. Let $\tau :H_{0}^{1}(\Omega )\to \mathbb{R}$ be as in
Lemma \ref{rep}. We write $x=(x',x_{N})\in \mathbb{R}^{N-1}\times
\mathbb{R\equiv R}^{N}$. For each $u\in H_{0}^{1}(\Omega )\subset H^{1}(%
\mathbb{R}^{N})$ and $0\leq s\leq 2$, define
\begin{equation*}
u_{s}(x):=\begin{cases}
[ (1-s)+s\tau (u)]u(x) &\hbox{if  }0\leq s\leq 1 \\
\tau (u)u(x',sx_{N}) &\hbox{if }x_{N}\geq 0,\hbox{ }1\leq s\leq 2 \\
\tau (u)u(x',x_{N}) &\hbox{if } x_{N}\leq 0,\hbox{ }1\leq s\leq 2
\end{cases}
\end{equation*}
Then $u_{s}\in H_{0}^{1}(\Omega )$, and
\begin{equation*}
\int_{\Omega }| \nabla u_{s}| ^2\leq s\int_{\Omega}| \nabla (\tau (u)u)| ^2,\quad
\int_{\Omega }| u_{s}| ^p\geq s^{-1}\int_{\Omega }| \tau (u)u| ^p,\quad
\mbox{if }1\leq s\leq 2.
\end{equation*}
Hence, for $1\leq s\leq 2$,
\begin{equation}
I(u_{s})\leq sI(u_{s})\leq \frac{s^2}{2}\int_{\Omega }| \nabla
(\tau (u)u)| ^2-\frac{1}{p}\int_{\Omega }| \tau
(u)u| ^p\leq I^{\#}(\tau (u)u).  \label{us}
\end{equation}%
Let $\Omega _{2}:=\{(x',\frac{1}{2}x_{N}):x\in \Omega $,
 $x_{N}\geq 0\}\cup \{x\in \Omega :x_{N}\leq 0\}$. By Lemma \ref{lem},
$\Omega_{2}\varsubsetneq \Omega $. Thus, we may choose
$\omega \in C_{c}^{\infty }(\Omega \setminus \Omega _{2})$, $\omega \neq 0$,
with
\begin{equation}
\int_{\Omega }| \nabla \omega | ^2=\int_{\Omega}| \omega | ^p.
\end{equation}
Define $\psi =(\psi ^{+},\psi ^{-}):X_{k}^n\times [ 0,\infty
)\to X^n$ by
\begin{gather*}
\psi ^{+}(z,s):=\begin{cases}
[ \varphi ^{+}(z)] _{s} & \hbox{if  }0\leq s\leq 2 \\
[ \varphi ^{+}(z)] _{2}+(s-2)\omega & \hbox{if  }2\leq s
\end{cases}\\
\psi ^{-}(z,s):=\begin{cases}
[ (1-s)+\alpha s]\varphi ^{-}(z) & \hbox{if  }0\leq s\leq 1 \\
\alpha \varphi ^{-}(z) & \hbox{if  }1\leq s
\end{cases}
\end{gather*}
with $\alpha :=2^{(3p-2)/(p-2)}$.  Then (i) holds. Lemma \ref{rep}, together
 with (\ref{us}), yields
\begin{equation*}
I(\psi ^{+}(z,s))\leq \begin{cases}
I(\varphi ^{+}(z)) &\hbox{if }0\leq s\leq 1 \\
I^{\#}(\tau (\varphi ^{+}(z))\varphi ^{+}(z)) & \hbox{if }1\leq s\leq 2 \\
I^{\#}(\tau (\varphi ^{+}(z))\varphi ^{+}(z))+I((s-2)\omega ) & \hbox{if  }
2\leq s\,.
\end{cases}
\end{equation*}
Indeed, the first inequality follows from Lemma \ref{rep}(i), the
second one is a consequence of (\ref{us}), and the third inequality follows
from the second one because $\omega $ and $u_{2}$ have disjoint supports for
every $u\in H_{0}^{1}(\Omega )$ and, therefore,
\begin{equation*}
I([ \varphi ^{+}(z)] _{2}+(s-2)w)=I([ \varphi ^{+}(z)]_{2})+I((s-2)w).
\end{equation*}
Clearly,
\begin{equation*}
J(\psi ^{-}(z,s))\geq \begin{cases}
J(\varphi ^{-}(z)) & \hbox{if  }0\leq s\leq 1 \\
\alpha J(\varphi ^{-}(z)) & \hbox{if  }1\leq s\,.
\end{cases}
\end{equation*}
Hence, Lemma \ref{rep} yields
\begin{equation}
\Phi _{0}(\psi (z,s))=I(\psi ^{+}(z,s))-J(\psi ^{-}(z,s))\leq \max \{\alpha
\Phi _{0}(\varphi (z))+\beta ,\hbox{ }0\},  \label{u}
\end{equation}
with $\beta :=I(\omega )$. Thus, (iii) holds. Finally, let $R'\geq \max \{2,R\}$
 be such that
\begin{gather*}
I(u)\leq -1,\quad \tau (u)=1,\quad I^{\#}(u)\leq -I(\omega )-1 \quad
\hbox{if }u\in X_{k}^{+},\;\Vert u\Vert \geq R', \\
I((s-2)\omega )\leq -\max \{I^{\#}(\tau (\varphi ^{+}(z))\varphi
^{+}(z)):z\in X_{k}^n\}-1 \quad \hbox{if }s\geq R', \\
J(v)\geq \max \{I(\psi ^{+}(z,s)):z\in X_{k}^n,\; s\geq 0\}+1 \quad
\hbox{if }v\in X_{n}^{-},\; \Vert v\Vert \geq R'.
\end{gather*}
Since, for $z=(u,v)$,
\begin{equation*}
\max \{\| u\| ,\| v\| \}\geq R'\Longrightarrow \| z\| \geq R
\Longrightarrow \varphi (z)=z,
\end{equation*}%
the previous inequalities yield
\begin{gather*}
\Phi _{0}(\psi (z,s)) \leq I(\psi ^{+}(z,s))\leq -1\quad \hbox{if }z=(u,v),
\; \max \{\| u\| ,s\}\geq R', \\
\Phi _{0}(\psi (z,s)) =I(\psi ^{+}(z,s))-J(\psi ^{-}(z,s))\leq -1
\quad\hbox{if }z=(u,v),\; \| v\| \geq R'.
\end{gather*}
This proves (ii).
\end{proof}

\begin{proposition} \label{H7}
The function $\Phi $ satisfies (H7). More precisely, there exist
$\alpha ,\beta >0$, depending only on $\Omega $ and $p$, with the following
property: For
every odd map $\sigma :X_{k}^n\to X^n$ such that $\sigma (z)=z$
if $\| z\| \geq R$, there exist an odd map $\widetilde{\sigma }
:X_{k+1}^n\to X^n$ and an $\widetilde{R}\geq R$ which satisfy:
\begin{itemize}
\item[(i)]  $\widetilde{\sigma }(z)=\sigma (z)$ \ if $z\in X_{k}^n$.

\item[(ii)]  $\widetilde{\sigma }(z)=z$ \ \ if $\ \| z\| \geq \widetilde{R}$.

\item[(iii)] $\Phi _{0}(\widetilde{\sigma }(z))\leq \alpha \Phi _{0}(\sigma (\pi
(z)))+\beta $ \ where $\pi :X_{k+1}^n\to X_{k}^n$ is the
orthogonal projection.
\end{itemize}
\end{proposition}

\begin{proof}
For $\varphi :=\sigma $ and $R$, let $\psi :X_{k}^n\times [ 0,\infty
)\to X^n$ and $R'>R$ be as in Lemma \ref{mainlem}. Fix
$e\in X_{k+1}^{+}$ orthogonal to $X_{k}^{+}$ with $\| e\| =1$, and
extend $\sigma $ to the half space $W=\{(z,se):z\in X_{k}^n$,
$s\geq 0\}\subset X_{k+1}^n$ by setting
\begin{equation*}
\sigma (z,se)=\psi (z,s)\quad \hbox{if }z\in X_{k}^n,\hbox{ }s\geq 0.
\end{equation*}
Then there exists $R''\geq R'$ such that $\Phi
_{0}(w)\leq -1$ and $\Phi _{0}(\sigma (w))\leq -1$ for all $w\in W$ with $%
\| w\| \geq R''$. The sublevel set
\begin{equation*}
D:=\{z\in X^n:\Phi _{0}(z)\leq -1\}
\end{equation*}
is homotopy equivalent to the unit sphere in $X^n$. Therefore, it is
contractible. Hence, there exists a homotopy
\begin{equation*}
\Psi :\{w\in W:\| w\| =R''\}\times [
0,1]\to D
\end{equation*}
such that $\Psi (w,0)=\sigma (w)$, $\Psi (w,1)=w$, and $\Psi (z,t)=z$ for $%
z\in X_{k}^n$, $t\in [ 0,1]$. Let $\widetilde{R}:=R''+1$ and define
\begin{equation*}
\widetilde{\sigma }(w)=\begin{cases}
\sigma (w) &\hbox{if }w\in W,\; \| w\| \leq R''  \\
\frac{\| w\| }{R''}\Psi (R''\frac{w }{\| w\| },\| w\| -R'') &\hbox{if  }
w\in W, \; R''\leq \| w\| \leq \widetilde{R}
\\
w & \hbox{if }w\in W,\; \widetilde{R}\leq \| w\| \\
-\widetilde{\sigma }(-w) & \hbox{if }-w\in W
\end{cases}
\end{equation*}
Since $\sigma $ is odd, $\widetilde{\sigma }$ is well defined and it is, by
definition, an odd extension of $\sigma $ to $X_{k+1}^n$ which satisfies $%
\widetilde{\sigma }(w)=w$ if $\| w\| \geq \widetilde{R}$. Note
that $su\in D$ if $u\in D$ and $s\geq 1$. Hence $\Phi _{0}(\widetilde{\sigma
}(w))\leq -1$ if $\| w\| \geq R''$ and, by Lemma \ref{mainlem},
\begin{equation*}
\Phi _{0}(\widetilde{\sigma }(z,se))=\Phi _{0}(\psi (z,| s|))\leq
\alpha \Phi _{0}(\sigma (z))+\beta \quad \hbox{if }\| \left(z,se\right) \| \leq R'',
\end{equation*}
as claimed.
\end{proof}

We now give some estimates for the values $c_{k}$ defined in (\ref{ck}). We
shall use the semiclassical inequality of Cwickel \cite{cw}, Lieb \cite{l}
and Rosenbljum \cite{ro} which states that, if $V\in L^{N/2}(\mathbb{R}^N)$,
$N\geq 3$, then the number of negative or zero eigenvalues counted with
multiplicity of the Schr\"{o}dinger operator $-\Delta +V$ on $L^2(\mathbb{R%
}^N)$ is bounded by
\begin{equation*}
C_0 \int_{\mathbb{R}^N}|V^-|^{N/2}
\end{equation*}
for some $C_0>0$.

\begin{lemma}
\label{lb}There exist positive constants $B_{1},B_{2}$ such that
\begin{equation*}
B_{1}k^{\nu }\leq c_{k}\leq B_{2}k^{\nu }
\end{equation*}
where $\nu :=\frac{2}{N}\frac{p}{p-2}$.
\end{lemma}

\begin{proof}
For every $n,k\geq 1$ there exists $z_{k}^n\in X^n$ such that
\begin{equation*}
\Phi _{0}^n(z_{k}^n)\leq c_{k}^n,\quad (\Phi _{0}^n)'(z_{k}^n)=0, \quad
\mu _{0}(z_{k}^n)\geq k+n,
\end{equation*}
where $\mu _{0}(z_{k}^n)$ denotes the Morse index plus the nullity of
$z_{k}^n$ \cite{t}. Thus $z_{k}^n=(u_{k}^n,0)$, and $u_{k}^n$ is a
critical point of the restriction of $\Phi _{0}$ to $X^{+}$ with Morse index
\begin{equation*}
\mu _{0}(u_{k}^n)\geq k.
\end{equation*}
The semiclassical inequality of Cwickel \cite{cw}, Lieb \cite{l}, and
Rosenbljum \cite{ro} gives
\begin{equation*}
k\leq \mu _{0}(u_{k}^n)\leq C_{1}\int_{\Omega }| u_{k}^n|
^{\theta }
\end{equation*}
with $\theta =\frac{N}{2}(p-2)$. Using H\"{o}lder's inequality we obtain
\begin{equation*}
B_{1}k^{\nu }\leq C_{2}\Big( \int_{\Omega }| u_{k}^n| ^{\theta
}\Big) ^{p/\theta }\leq \frac{(p-2)}{2p}| u_{k}^n|
_{p}^p=\Phi _{0}^n(z_{k}^n)\leq c_{k}^n
\end{equation*}
with $\nu =\frac{p}{\theta }=\frac{2}{N}\frac{p}{p-2}$. This yields the
first inequality of this lemma. We turn to the second one. Let
\begin{equation*}
\Gamma _{k}^{+}:=\{\sigma ^{+}\in C^{0}(X_{k}^{+},X^{+}):\sigma ^{+}
\hbox{is odd, and $\sigma ^{+}(u)=u$  if $\| u\|$
is large enough} \}.
\end{equation*}
If $\sigma ^{+}\in \Gamma _{k}^{+}$ and $\sigma \in C^{0}(X_{k}^n,X^n)$
is given by $\sigma (u,v)=(\sigma ^{+}(u),v)$, then $\Phi _{0}(\sigma
(z))\leq 0$ for $\| z\| $ large enough. Arguing as in the proof
of Proposition \ref{H7} we may assume that $\sigma \in \Gamma _{k}^n$, and
that
\begin{equation*}
\max_{z\in X_{k}^n}\Phi _{0}(\sigma (z))=\max_{u\in X_{k}^{+}}\Phi
_{0}(\sigma ^{+}(u)).
\end{equation*}
Hence, inequality (40) in \cite{bl} yield
\begin{equation*}
c_{k}^n\leq \inf_{\sigma ^{+}\in \Gamma _{k}^{+}}\hbox{ }\max_{u\in
X_{k}^{+}}\Phi _{0}(\sigma ^{+}(u))\leq B_{2}k^{\nu }.
\end{equation*}
for all $n$. Our claim follows.
\end{proof}


\begin{proof}[Proof of Theorem \ref{homo}]
We have shown that assumptions (H1)-(H7) of Theorem \ref{abs} hold.
We now show that the sequence (\ref{seq}) is unbounded. Assume, by contradiction,
there exists a $B>0$ such that
\begin{equation*}
c_{k+1}-c_{k}\leq B(\theta (c_{k+1})+\theta (c_{k})+1)
\end{equation*}
where $\theta (s)=-\theta _{1}(t,s)=\theta _{2}(t,s)=A(s^2+1)^{\gamma /2}$.
 As in \cite[(10.47)]{r}, this implies the existence of a constant $D>0$
such that
\begin{equation*}
c_{k}\leq Dk^{1/(1-\gamma )}\quad\hbox{for all }k.
\end{equation*}
This contradicts Lemma \ref{lb} if $\gamma <( \frac{1}{p}-\frac{1 }{2^{\ast }}) N$,
 as assumed. Thus, Theorem \ref{abs}\ yields a sequence
of critical values $(\widetilde{c}_{k})$ of $\Phi _{1}$ which satisfy
\begin{equation}
\zeta _{2}(1,c_{k})<\widetilde{c}_{k}\leq \zeta _{2}(1,\alpha
(c_{k}+1)+\beta )  \label{ref}
\end{equation}
with $a,\beta >0$ as in Proposition \ref{H7} (see Remark \textsl{(f)} at the
end of section 2). For our particular $\theta $, we have that $s\leq \zeta
_{2}(1,s)\leq A_{1}(s+1)$. Thus, (\ref{ref}) and Lemma \ref{lb} yield
\begin{equation*}
C_{1}k^{\nu }\leq \widetilde{c}_{k}\leq C_{2}k^{\nu }
\end{equation*}
with $\nu :=\frac{2p}{N(p-2)}$, as claimed.
\end{proof}

\begin{proof}[Proof of Theorem \ref{intro}]
Assume that $f$ satisfies the assumptions of Theorem \ref{intro}.
We show that it satisfies also those of
Theorem \ref{homo}, with the same $\gamma $.
Since $0\leq \sigma -1\leq \frac{q}{p}(\gamma p-1)$ and $p\leq q$,
\begin{equation*}
|f_{z}(x,u,v)|\leq d(|u|^{\gamma p-1}+|v|^{\gamma q-1}+1).
\end{equation*}%
Using, in addition, Young's inequality and the fact that
$\gamma \geq \frac{1}{p}$ we obtain
\begin{align*}
|f_{z}(x,z)z| &\leq d_{1}(|u|^{\gamma p}+|v|^{\gamma q}+|u|^{\gamma
p-1}|v|+|u||v|^{\sigma -1}+1) \\
&\leq d(|u|^{\gamma p}+|v|^{\gamma q}+1).
\end{align*}%
Moreover,
\begin{align*}
|f(x,u,v)|-|f(x,0,0)| &\leq |f(x,u,v)-f(x,u,0)|+|f(x,u,0)-f(x,0,0)| \\
&\leq \int_{0}^{|v|}|f_{v}(x,u,\xi )|d\xi +\int_{0}^{|u|}|f_{u}(x,\xi
,0)|d\xi \\
&\leq d_{2}(|u|^{\gamma p-1}|v|+|u|^{\gamma p}+|v|^{\gamma q}+1) \\
&\leq d(|u|^{\gamma p}+|v|^{\gamma q}+1)
\end{align*}
Thus, $f$ satisfies the assumptions of Theorem \ref{homo}.
\end{proof}

In particular, if $f(x,u,v)=g(x)u$  and $p<\frac{2N-2}{N-2}$,
Theorem \ref{homo} yields infinitely many solutions of the perturbed
elliptic equation
\begin{gather*}
-\Delta u=| u| ^{p-2}u+g(x)\quad \hbox{in }\Omega \\
u=0\quad \hbox{on }\partial \Omega ,
\end{gather*}
which is Bahri and Lions's result \cite{bl}. The upper estimates for their
energy were recently established by Castro and Clapp \cite{cc}.

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