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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 105, 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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/105\hfil Existence of solutions]
{Existence of solutions for p-Kirchhoff
type problems with critical exponent}

\author[A. Hamydy, M. Massar, N. Tsouli\hfil EJDE-2011/105\hfilneg]
{Ahmed Hamydy, Mohammed Massar, Najib Tsouli}  % in alphabetical order

\address{Ahmed Hamydy \newline
University Mohamed I, Faculty of sciences,
Department of Mathematics, Oujda, Morocco}
\email{a.hamydy@yahoo.fr}

\address{Mohammed Massar \newline
University Mohamed I, Faculty of sciences,
Department of Mathematics, Oujda, Morocco}
\email{massarmed@hotmail.com}

\address{Najib Tsouli \newline
University Mohamed I, Faculty of sciences,
Department of Mathematics, Oujda, Morocco}
\email{tsouli@hotmail.com}

\thanks{Submitted July 26, 2011. Published August 16, 2011.}
\subjclass[2000]{35A15, 35B33, 35J62}
\keywords{p-Kirchhoff; critical exponent; parameter; Lions principle}

\begin{abstract}
 We study the  existence of solutions for the
 p-Kirchhoff type problem involving the critical Sobolev
 exponent,
 \begin{gather*}
 -\Big[g\Big(\int_\Omega|\nabla u|^pdx\Big)\Big]\Delta_pu
 =\lambda f(x,u)+|u|^{p^\star-2}u\quad\text{in }\Omega,\\
 u=0\quad\text{on }\partial\Omega,
 \end{gather*}
 where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$,
 $1<p<N$, $p^\star=Np/(N-p)$ is  the  critical  Sobolev
 exponent, $\lambda$ is a positive parameter, $f$ and $g$ are
 continuous functions. The main results of this paper establish, via
 the variational method. The concentration-compactness principle
 allows to prove that the Palais-Smale condition is satisfied below a
 certain level.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction and main results}

We are concerned with the existence of solutions for
the  p-Kirchhoff type problem
\begin{equation}\label{E11}
\begin{gathered}
-\Big[g\Big(\int_\Omega|\nabla u|^pdx\Big)\Big]\Delta_pu
 =\lambda f(x,u)+|u|^{p^\star-2}u \quad \text{in }\Omega\\
u=0\quad \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$,
$1<p<N$, $p^\star=Np/(N-p)$ is  the  critical  Sobolev
exponent, and
$f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$,
$g:\mathbb{R}^+\to\mathbb{R}^+$ are continuous functions that
satisfy the following conditions:
\begin{itemize}

\item[(F1)]
$f(x,t)=o(|t|^{p-1})$ as $t\to0$, uniformly for $x\in\Omega$;

\item[(F2)] There exists $q\in(p,p^\star)$ such that
$$
\lim_{|t|\to+\infty} \frac{f(x,t)}{|t|^{q-2}t}=0,\quad
\text{uniformly for} x\in\Omega.
$$
\item[(F3)] There exists $\theta\in(p/\sigma,p^\star)$
such that
$0<\theta F(x,t)\leq tf(x,t)$ for all $x\in\Omega$
and $t\neq0$,
where $F(x,t)=\int_0^tf(x,s)ds$ and $\sigma$ is given by (G2) below.

\item[(G1)] There exists $\alpha_0>0$ such that
$g(t)\geq\alpha_0$ for all $t\geq0$;

\item[(G2)] There exists $\sigma>p/p^\star$ such that
$G(t)\geq\sigma g(t)t$ for all $t\geq0$,
where $G(t)=\int_0^tg(s)ds$;

\end{itemize}

Much interest has grown on problems involving critical
exponents, starting from the celebrated paper by Brezis and
Nirenberg \cite{BN}, where the case $p=2$ is  considered. We refer
the reader to \cite{AA,DH,FF} and reference therein for the study of
problems with critical exponent.


Problem \eqref{E11} is a general version of a model presented by
Kirchhoff \cite{G}. More precisely, Kirchhoff introduced a model
\begin{equation}\label{E12}
\rho\frac{\partial^2u}{\partial
t^2}-\Big(\frac{\rho_0}h+\frac{E}{2L}\int_0^L|\frac{\partial
u}{\partial x}|^2dx\Big)\frac{\partial^2u}{\partial x^2}=0,
\end{equation}
where $\rho, \rho_0, h, E, L$ are constants, which extends the
classical D'Alembert's wave equation by considering the effects of
the changes in the length of the strings during the vibrations. The
 problem
\begin{equation}\label{E13}
\begin{gathered}
-\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta
u=f(x,u)\quad \text{in }\Omega\\
u=0\quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
received much attention, mainly after the article by Lions \cite{JL}.
Problems like \eqref{E13} are also introduced as
models for other physical phenomena as, for example, biological
systems where $u$ describes a process which depends on the average
of itself (for example, population density). See \cite{ACM} and its
references therein.
For a more detailed reference on this subject we refer the
interested reader to \cite{AP,CCS,CF,CN,M,PZ}.

Motivated by the ideas in \cite{ACF}, our approach for studying 
problem \eqref{E11} is variational and uses
minimax critical point theorems. The difficulty is due to the lack
of compactness of the imbedding $W_0^{1,p}(\Omega)\hookrightarrow
L^{p^{\star}}(\Omega)$ and the Palais-Smale condition for the
corresponding energy functional could not be checked directly. So
the concentration-compact principle of  Lions \cite{PL} is
applied to deal with this difficulty.

The main result of this paper is the following theorem.

\begin{theorem}\label{theo11}
Suppose that {\rm (G1)--(G2), (F1)--(F3)} hold. Then, there exists
$\lambda_*>0$, such that \eqref{E11} has a nontrivial
solution for all $\lambda\geq\lambda_*$.
\end{theorem}

\section{Preliminary results}

We consider the  energy functional $I:
W_0^{1,p}(\Omega)\to\mathbb{R}$ defined by
\begin{equation}\label{E21}
I(u)=\frac1pG(\|u\|^p)-\lambda\int_\Omega
F(x,u)dx-\frac1{p^\star}\int_\Omega |u|^{p^\star}dx,
\end{equation}
where  $W_0^{1,p}(\Omega)$ is the Sobolev space endowed with the
norm $\|u\|^p=\int_\Omega|\nabla u|^pdx$. It is well known that a
critical point of $I$ is a weak solution of problem \eqref{E11}.

To use variational methods, we give some results related to
the Palais-Smale compactness condition.
Recall that a sequence $(u_n)$ is a Palais-Smale  sequence of $I$ at
the level $c$, if $I(u_n)\to c$ and $I'(u_n)\to0$.

In the sequel, we show that the functional $I$ has the mountain pass
geometry. This purpose is proved in the next lemmas.

\begin{lemma}\label{lem21}
Suppose that {\rm (F1), (F2), (G1)} hold.
Then, there exist $r,\rho>0$ such that
$\inf_{\|u\|=r}I(u)\geq\rho>0$.
\end{lemma}

\begin{proof}
It follows from (F1) and (F2) that for any
$\varepsilon>0$, there exists $C(\varepsilon)>0$.
\begin{equation}\label{E22}
F(x,t)\leq\frac1p\varepsilon |t|^p+C(\varepsilon)|t|^q\quad
\text{for all }t.
\end{equation}
By (G1) and the Sobolev embdding, we have
\begin{equation} \label{E23}
\begin{split}
I(u)&\geq \frac{\alpha_0}{p}\|u\|^p-\lambda
C_1\varepsilon\|u\|^p-\lambda
C_2(\varepsilon)\|u\|^q-C_3\|u\|^{p^\star}\\
&=\|u\|\Big[\big(\frac{\alpha_0}{p}-\lambda
C_1\varepsilon\big)\|u\|^{p-1}-\lambda
C_2(\varepsilon)\|u\|^{q-1}-C_3\|u\|^{p^\star-1}\Big].
\end{split}
\end{equation}
Taking $\varepsilon=\alpha_0/(2p\lambda C_1)$ and setting
$$
\xi(t)=\frac{\alpha_0}{2p}t^{p-1}-\lambda
C_2t^{q-1}-C_3t^{p^\star-1}.
$$ Since $p<q<p^\star$, we see that there
exist $r>0$ such that $\underset{t\geq0}\max \xi(t)=\xi(r)$. Then,
by \eqref{E23}, there exists $\rho>0$ such that $I(u)\geq\rho$ for
all $\|u\|=r$.
\end{proof}

\begin{lemma}\label{lem22}
Suppose that {\rm (G2), (F3)} hold. Then for
all $\lambda>0$, there exists a nonnegative function
$e\in W_0^{1,p}(\Omega)$ independent of $\lambda$, such that
$\|e\|>r$ and $I(e)<0$.
\end{lemma}

\begin{proof}
Choose a nonnegative function $\phi_0\in C_0^\infty(\Omega)$ with
$\|\phi_0\|=1$. By integrating (G2), we obtain
\begin{equation}\label{E24}
G(t)\leq \frac{G(t_0)}{t_0^{1/\sigma}}t^{1/\sigma}
=C_0t^{1/\sigma}\quad \text{for all }t\geq t_0>0.
\end{equation}
By (F3),
$\int_\Omega F(x,t\phi_0)dx\geq0$.
Hence
$$
I(t\phi_0)\leq\frac{C_0}pt^{p/\sigma}
-\frac{t^{p^\star}}{p^\star}\int_\Omega
\phi_0^{p^\star}dx\quad \text{for all } t\geq t_0.
$$
Since $p/\sigma<p^\star$, the lemma is proved by choosing
$e=t_*\phi_0$ with $t_*>0$ large enough.
\end{proof}

In view of Lemmas \ref{lem21} and \ref{lem22}, we may apply a
version of the Mountain Pass theorem without Palais-Smale condition
to obtain a sequence $(u_n)\subset W_0^{1,p}(\Omega)$ such that
$$
I(u_n)\to c_*\quad\text{and}\quad I'(u_n)\to 0,
$$
where
\begin{equation}\label{E25}
c_*=\inf_{\gamma\in\Gamma} \max_{t\in[0,1]} I(\gamma(t))>0,
\end{equation}
with
$$
\Gamma=\big\{\gamma\in C([0,1],W_0^{1,p}(\Omega)):
\gamma(0)=0,\,I(\gamma(1))<0\big\}.
$$
Denoted by $S_*$ the best positive constant of the Sobolev embedding
$W_0^{1,p}(\Omega)\hookrightarrow L^{p^\star}(\Omega)$ given by
\begin{equation}\label{E26}
S_*=\inf\big\{\int_\Omega|\nabla u|^pdx: u\in
W_0^{1,p}(\Omega),\;\int_\Omega|u|^{p^\star}dx=1\big\}.
\end{equation}

\begin{lemma}\label{lem23}
Suppose that {\rm (G1)--(G2), (F1)--(F3)} hold. Then there exists
$\lambda_*>0$ such that $c_*\in
\big(0,(\frac1\theta-\frac1{p^\star})(\alpha_0
S_*)^{\frac{N}{p}}\big)$ for all $\lambda\geq\lambda_*$, where
$c_*$ is given by \eqref{E25}.
\end{lemma}

\begin{proof}
For $e$ given by Lemma \ref{E22}, we have
$\lim_{t\to+\infty} I(te)=-\infty$, then there
exists $t_\lambda>0$ such that
$I(t_\lambda e)=\underset{t\geq0}\max I(te)$.
Therefore,
$$
t_\lambda^{p-1}g(\|t_\lambda e\|^p)\|e\|^p
=\lambda \int_\Omega f(x,t_\lambda e)e\,dx
+t_\lambda^{p^\star-1}\int_\Omega e^{p^\star}dx;
$$
thus
\begin{equation}\label{E27}
g(\|t_\lambda e\|^p)\|t_\lambda e\|^p
=\lambda t_\lambda \int_\Omega f(x,t_\lambda e)e\,dx
+t_\lambda^{p^\star}\int_\Omega e^{p^\star}dx.
\end{equation}
By \eqref{E24}, it follows that
$$
\frac{C_0}\sigma\|e\|^{p/\sigma}t_\lambda^{p/\sigma}
\geq t_\lambda^{p^\star}\int_\Omega e^{p^\star}dx,
\quad \text{with } t_0<t_\lambda.
$$
Since $p/\sigma <p^\star$, $(t_\lambda)$ is bounded. So,
there exists a sequence $\lambda_n\to+\infty$ and $s_0\geq0$
such that $t_{\lambda_n}\to s_0$ as $n\to\infty$.
Hence, there exists $C>0$ such that
$$
g(\|t_{\lambda_n}e\|^p)\|t_{\lambda_n}e\|^p\leq C\quad
\text{for all }n;
$$
that is,
$$
\lambda_n t_{\lambda_n}\int_\Omega f(x,t_{\lambda_n} e)e\,dx
+t_{\lambda_n}^{p^\star}\int_\Omega e^{p^\star}dx\leq
C\quad \text{for all } n.
$$
If $s_0>0$, the above inequality implies that
$$
\lambda_n t_{\lambda_n}\int_\Omega f(x,t_{\lambda_n} e)e\,dx
+t_{\lambda_n}^{p^\star}\int_\Omega e^{p^\star}dx\to+\infty
\leq C,\quad \text{as } n\to\infty,
$$
which is impossible, and consequently $s_0=0$.
Let $\gamma_*(t)=te$. Clearly $\gamma_*\in\Gamma$, thus
$$
0<c_*\leq\max_{t\geq0} I(\gamma_*(t))
=I(t_\lambda e)\leq\frac1pG(\|t_\lambda e\|^p).
$$
Since $t_{\lambda_n}\to0$ and
$(\frac1\theta-\frac1{p^\star})(\alpha_0
S_*)^{N/p}>0$, for $\lambda>0$ sufficiently large, we have
$$
\frac1pG(\|t_\lambda
e\|^p)<\big(\frac1\theta-\frac1{p^\star}\big)(\alpha_0
S_*)^{N/p},
$$
and hence
$$
0<c_*<\big(\frac1\theta-\frac1{p^\star}\big)(\alpha_0
S_*)^{N/p}.
$$
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theo11}]
From Lemmas \ref{E21}, \ref{E22} and \ref{E23}, there exists a
sequence $(u_n)\subset W_0^{1,p}(\Omega)$ such that
\begin{equation}\label{E28}
I(u_n)\to c_*\quad \text{and}\quad I'(u_n)\to 0,
\end{equation}
with
$c_*\in\big(0,(\frac1\theta-\frac1{p^\star})(\alpha_0
S_*)^{N/p}\big)$ for $\lambda\geq\lambda_*$. Then, there
exists $C>0$ such that $|I(u_n)|\leq C$, and by (F3) for
$n$ large enough, it follows from (G1) and (G2) that
\begin{equation} \label{E29}
\begin{split}
C+\|u_n\|
&\geq I(u_n)-\frac1\theta\langle I'(u_n),u_n\rangle\\
&\geq \frac1pG(\|u\|^p)-\frac1\theta
g(\|u_n\|^p)\|u_n\|^p\\
&\geq \big(\frac\sigma{p}-\frac1\theta\big)\alpha_0\|u_n\|^p.
\end{split}
\end{equation}
Since $\theta>p/\sigma$, $(u_n)$ is bounded. Hence, up to a
subsequence, we may assume that
\begin{equation} \label{E210}
\begin{gathered}
u_n\rightharpoonup u \quad \text{weakly in } W_0^{1,p}(\Omega),\\
u_n\to u\quad \text{a.e. in }\Omega,\\
u_n\to u\quad \text{in }L^s(\Omega),\;1\leq s<p^\star,\\
|\nabla u_n|^p\rightharpoonup\mu\quad (\text{weak*-sense of measures })\\
|u_n|^{p^\star}\rightharpoonup\nu\quad
(\text{weak*-sense of measures}),
\end{gathered}
\end{equation}
where $\mu$ and $\nu$ are a nonnegative  bounded measures on
$\overline{\Omega}$. Then, by concentration-compactness principle
due to Lions \cite{PL}, there exists some at most countable index
set $J$ such that
\begin{equation}\label{E211}
\begin{gathered}
\nu=|u|^{p^\star}+\underset{j\in
J}\sum\nu_j\delta_{x_j}\,,\quad \nu_j>0,\\
\mu\geq|\nabla u|^p+\sum_{j\in J} \mu_j\delta_{x_j}\,,\quad \mu_j>0,\\
S_*\nu_j^{p/p^\star}\leq\mu_j,
\end{gathered}
\end{equation}
where $\delta_{x_j}$ is the Dirac measure mass at
$x_j\in\overline{\Omega}$.

Let $\psi(x)\in C_0^\infty$ such that $0\leq\psi\leq1$,
\begin{equation}\label{E212}
\psi(x)=\begin{cases}
1&\text{if }|x|<1\\
0&\text{if }|x|\geq2
\end{cases}
\end{equation}
and $|\nabla\psi|_\infty\leq 2$.

For $\varepsilon>0$ and $j\in J$, denote
$\psi_\varepsilon^j(x)=\psi((x-x_j)/\varepsilon)$. Since
$I'(u_n)\to 0$ and $(\psi_\varepsilon^ju_n)$ is bounded,
$\langle I'(u_n),\psi_\varepsilon^ju_n\rangle\to0$ as
$n\to\infty$; that is,
\begin{equation} \label{E213}
\begin{split}
&g(\|u_n\|^p)\int_\Omega|\nabla u_n|^p\psi_\varepsilon^jdx\\
&=-g(\|u_n\|^p)\int_\Omega u_n|\nabla u_n|^{p-2}\nabla
u_n\nabla\psi_\varepsilon^jdx\\
&\quad +\lambda\int_\Omega
f(x,u_n)u_n\psi_\varepsilon^jdx
+\int_\Omega|u_n|^{p^\star}\psi_\varepsilon^jdx+o_n(1).
\end{split}
\end{equation}
By \eqref{E210} and Vitali's theorem, we see that
$$
\lim_{n\to\infty} \int_{\Omega}|u_n\nabla\psi_\varepsilon^j|^p dx
=\int_{\Omega}|u\nabla\psi_\varepsilon^j|^p dx
$$
Hence, by  H\"{o}lder's inequality we obtain
\begin{equation}\label{E214}
\begin{split}
&\limsup_{n\to\infty} \big|\int_\Omega u_n|\nabla
u_n|^{p-2}\nabla u_n\nabla\psi_\varepsilon^jdx\big|\\
&\leq \limsup_{n\to\infty} \Big(\int_{\Omega}|\nabla
u_n|^pdx\Big)^{(p-1)/p}
\Big(\int_{\Omega}|u_n\nabla\psi_\varepsilon^j|^p
dx\Big)^{1/p}\\
&\leq C_1\Big(\int_{B(x_j,2\varepsilon)}|u|^p|\nabla
\psi_\varepsilon^j|^p dx\Big)^{1/p}\\
&\leq C_1\Big(\int_{B(x_j,2\varepsilon)}|\nabla
\psi_\varepsilon^j|^Ndx\Big)^{1/N}
\Big(\int_{B(x_j,2\varepsilon)}|u|^{p^\star} dx\Big)^{1/p^\star}\\
&\leq C_2\Big(\int_{B(x_j,2\varepsilon)}|u|^{p^\star}
dx\Big)^{1/p^\star} \to 0\quad \text{as } \varepsilon\to0\,.
\end{split}
\end{equation}

On the other hand, from \eqref{E210} we have
$$
f(x,u_n)u_n\to f(x,u)u\quad \text{a.e. in }\Omega,
$$
and $u_n\to u$ strongly in $L^p(\Omega)$ and in $L^q(\Omega)$.
By (F1)--(F3), for any $\varepsilon>0$ there
exists $C_\varepsilon>0$ such that
\begin{equation}\label{E215}
|f(x,t)|\leq\varepsilon|t|^{p-1}+C_\varepsilon|t|^{q-1}\quad
\text{for all }(x,t)\in\Omega\times\mathbb{R};
\end{equation}
thus
$$
|f(x,u_n)u_n|\leq\varepsilon|u_n|^p+C_\varepsilon|u_n|^q.
$$
This is what we need to apply Vitali's theorem, which yields
$$
\lim_{n\to\infty} \int_\Omega
f(x,u_n)u_ndx=\int_\Omega f(x,u)u\,dx.
$$
Since $\psi_\varepsilon^j$ has compact support, letting
$n\to\infty$ in \eqref{E213} we deduce  from \eqref{E210}
and \eqref{E214} that
$$
\alpha_0\int_\Omega\psi_\varepsilon^jd\mu\leq
C_2\Big(\int_{B(x_j,2\varepsilon)}|u|^{p^\star}
dx\Big)^{1/ p^\star} +\lambda\int_{B(x_j,2\varepsilon)}
f(x,u)udx+\int_\Omega\psi_\varepsilon^jd\nu.
$$
Letting $\varepsilon\to0$, we obtain
$\alpha_0\mu_j\leq\nu_j$.
Therefore,
\begin{equation}\label{E216}
(\alpha_0 S_*)^{N/p}\leq\nu_j.
\end{equation}
We will prove that this inequality is not possible.
Let us assume that
$(\alpha_0 S_*)^{N/p}\leq\nu_{j_0}$ for some $j_0\in J$. From
(G2) we see that
$$
\frac1pG(\|u_n\|^p)-\frac1\theta
g(\|u_n\|^p)\|u_n\|^p\geq0\quad \text{for all }n.
$$
Since
$$
c_*=I(u_n)-\frac1\theta \langle I'(u_n),u_n\rangle+o_n(1),
$$
it follows that
\begin{align*}
c_*&\geq \big(\frac1\theta-\frac1{p^\star}\big)
\int_\Omega|u_n|^{p^\star}dx+o_n(1)\\
&\geq \big(\frac1\theta-\frac1{p^\star}\big)
\int_\Omega\psi_\varepsilon^{j_0}|u_n|^{p^\star}dx+o_n(1)
\end{align*}
Letting $n\to\infty$, we obtain
\[
c_*\geq \big(\frac1\theta-\frac1{p^\star}\big)\sum_{j\in
J}\psi_\varepsilon^{j_0}(x_j)\nu_j
\geq \big(\frac1\theta-\frac1{p^\star}\big)(\alpha_0 S_*)^{N/p}.
\]
This contradicts Lemma \ref{lem23}. Then $J=\emptyset$, and
hence $u_n\to u$ in $L^{p^\star}(\Omega)$. By \eqref{E215}
we have
\begin{align*}
\int_\Omega|f(x,u_n)(u_n-u)|dx
&\leq \int_\Omega\big(\varepsilon|u_n|^{p-1}
+C_\varepsilon|u_n|^{q-1}\big)|u_n-u|dx\\
&\leq \varepsilon \Big(\int_\Omega|u_n|^pdx\Big)^{p-1)/p}
\Big(\int_\Omega|u_n-u|^pdx\Big)^{1/p}\\
&\quad +C_\varepsilon\Big(\int_\Omega|u_n|^qdx\Big)^{(q-1)/q}
\Big(\int_\Omega|u_n-u|^qdx\Big)^{1/q}.
\end{align*}
Then, using again \eqref{E210}, we obtain
\begin{equation}\label{E217}
\lim_{n\to\infty} \int_\Omega f(x,u_n)(u_n-u)dx=0.
\end{equation}
Since $u_n\to u$ in $L^{p^\star}(\Omega)$, we see that
\begin{equation}\label{E218}
\lim_{n\to\infty} \int_\Omega|u_n|^{p^\star-2}u_n(u_n-u)dx=0.
\end{equation}
 From $\langle I'(u_n),u_n-u\rangle=o_n(1)$, we deduce that
\begin{align*}
\langle I'(u_n),u_n-u\rangle
&= g(\|u_n\|^p)\int_\Omega|\nabla
u_n|^{p-2}\nabla u_n\nabla(u_n-u)dx\\
&\quad -\lambda\int_\Omega
f(x,u_n)(u_n-u)dx-\int_\Omega|u_n|^{p^\star-2}u_n(u_n-u)dx
=o_n(1)
\end{align*}
This, \eqref{E217} and \eqref{E218} imply
$$
\lim_{n\to\infty}  g(\|u_n\|^p)\int_\Omega|\nabla
u_n|^{p-2}\nabla u_n\nabla(u_n-u)dx=0.
$$
Since $u_n$ is bounded and $g$ is continuous, up to subsequence,
there is $t_0\geq0$ such that
$$
g(\|u_n\|^p)\to
g(t_0^p)\geq\alpha_0,\quad \text{as } n\to\infty,
$$
and so
$$
\underset{n\to\infty}\lim\int_\Omega|\nabla
u_n|^{p-2}\nabla u_n\nabla(u_n-u)dx=0.
$$
Thus by the $(S_+)$ property, $u_n\to u$ strongly in
$W_0^{1,p}(\Omega)$, and hence $I'(u)=0$. The proof is complete.
\end{proof}

\section{A special case}
We consider the  problem
\begin{equation}\label{E31}
\begin{gathered}
-\Big(\alpha+\beta\int_\Omega|\nabla u|^pdx\Big)\Delta_p u
=\lambda f(x,u)+|u|^{p^\star-2}u\quad \text{in }\Omega\\
u=0\quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded smooth domain of
$\mathbb{R}^N$, $1<p<N<2p$, $\alpha$ and $\beta$ are a positive
constants.

Set $g(t)=\alpha+\beta t$. Then, $g(t)\geq\alpha$ and
$$
G(t)=\int_0^1g(s)ds=\alpha t+\frac12\beta
t^2\geq\frac12(\alpha+\beta t)t=\sigma g(t)t
$$
where $\sigma=1/2$. Hence the conditions (G1) and
(G2) are satisfied.

For this case, a typical example of a function satisfying the
conditions (F1)--(F3) is given by
$$
f(x,t)=\sum_{i=1}^ka_i(x)|t|^{q_i-2}t,
$$
where $k\geq1$, $2p<q_i<p^\star$ and
$a_i(x)\in C(\overline{\Omega})$.
In view of Theorem \ref{theo11}, we have the following corollary.

\begin{corollary}
Suppose that {\rm (F1)--(F3)} hold. Then, there
exists $\lambda_*>0$, such that problem \eqref{E31} has a
nontrivial solution for all $\lambda\geq\lambda_*$.
\end{corollary}

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\end{document}


\eqref{E11}

\eqref{E11}