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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 69, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/69\hfil Nonhomogeneous Kirchhoff equations]
{Nonhomogeneous elliptic problems of Kirchhoff type involving critical
Sobolev exponents}

\author[S. Benmansour, M. Bouchekif \hfil EJDE-2015/69\hfilneg]
{Safia Benmansour, Mohammed Bouchekif}

\address{Safia Benmansour \newline
Laboratoire des Syst\`emes Dynamiques et Applications.
 Facult\'e des Sciences, Universit\'e de Tlemcen. BP 119, 13000 Tlemcen,
 Alg\'erie}
\email{safiabenmansour@hotmail.fr}

\address{Mohammed Bouchekif \newline
Laboratoire des Syst\`emes Dynamiques et Applications.
Facult\'e des Sciences, Universit\'e de Tlemcen. BP 119, 13000 Tlemcen,
 Alg\'erie}
\email{m\_bouchekif@yahoo.fr}

\thanks{Submitted December 10, 2014. Published March 24, 2015.}
\subjclass[2000]{35J20, 35J60, 47J30}
\keywords{Variational methods; critical Sobolev exponent;  Nehari manifold;
 \hfill\break\indent Palais-Smale condition; Kirchhoff equation}

\begin{abstract}
 This article concerns  the existence and the multiplicity of solutions
 for nonhomogeneous elliptic Kirchhoff problems involving the critical
 Sobolev exponent, defined on a regular bounded domain of
 $\mathbb{R}^3$. Our approach is essentially based on Ekeland's
 Variational Principle and the Mountain Pass Lemma.
\end{abstract}

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

\section{Introduction}

In this work we study the existence and the multiplicity of solutions for
the  problem
\begin{equation} \label{eP}
\begin{gathered}
-(a\int_{\Omega }|\nabla u|^2dx+b)\Delta u=| u| ^4u+f
\quad \text{in }\Omega , \\
u=0\quad \text{on }\partial \Omega ,
\end{gathered}
\end{equation}
where $\Omega $ is a smooth bounded domain of $\mathbb{R}^3$, 
$a,b$ are positive constants and $f$ belongs to 
$H^{-1}$ (the topological dual of $H_0^1(\Omega )) $
satisfying suitable conditions.

The original one-dimensional Kirchhoff equation was introduced by Kirchhoff
\cite{K} in 1883. His model takes into account the changes in length
of the strings produced by transverse vibrations.

Problem \eqref{eP} is called nonlocal because of the presence of
the integral over the entire domain $\Omega $, which implies that the
equation in \eqref{eP} is no longer a pointwise identity.

Problem \eqref{eP} is related to the stationary analog of the Kirchhoff equation
\begin{gather*}
u_{tt}-(a\int_{\Omega }| \nabla u|
^2dx+b)\bigtriangleup u=h(x,u) \quad \text{in }
\Omega \times (0,\text{ }T), \\
u=0 \quad \text{in }\partial \Omega \times (0,\text{ }T), \\
u(x,0)=u_0(x),\quad u_{t}(x,0)=u_1(x),
\end{gather*}
where $T$ is a positive constant, $u_0$ and $u_1$ are given functions.
It can be seen as a generalization of the classical D'Alembert wave equation
for free vibrations of elastic strings. For such problems, $u$ denotes the
displacement, $h(x$, $u)$ the external force, $b$ is the initial tension and
$a$ is related to the intrinsic properties of the strings (such as Young's
modulus). For more details, we refer the readers to the work of D'Ancona and
Shibata \cite{AS} and the references therein.

Nonlocal problems arise not only from mathematical and physical fields but
also from several other branches. When they appear in biological systems, $u$
describes a process depending on the average of itself, as population
density. Their theoretical study has attracted a lot of interests from
mathematicians for a long time and many works have been done. We quote in
particular the famous article of Lions \cite{L}. However in most of papers,
the used approach relies on topological methods.

In the last two decades, many authors have considered the stationary
elliptic problem
\begin{equation} \label{ePS}
\begin{gathered}
-\Big(a\int_{\Omega }|\nabla u|^2dx+b\Big)\Delta u=h(x,u) \quad
 \text{in }\Omega \\
u=0 \quad \text{on }\partial \Omega ,
\end{gathered}
\end{equation}
where $\Omega \subset\mathbb{R}^{N}$ and $h(x,u)$ is a continuous function, 
see for example \cite{ACM}.
Alves and colleagues were the first to obtain existence results via
variational methods. After this breakthrough, many works have been done in
this direction. One can quote \cite{BB} for the case where $h(x,u)$ is
asymptotically linear at infinity.

Problem \eqref{ePS} has also been extensively studied in the
whole space when the potential function has a subcritical or critical
growth, for more details see \cite{LS}.

In the case of a bounded domain of $\mathbb{R}^{N}$ with $N\geq 3$, 
Tarantello \cite{T}  proved, under a suitable
condition on $f$, the existence of at least two solutions to 
\eqref{ePS} for $a=0$, $b=1$ and $h(x,u)=|u|^{4/(N-2)}u+f$.

A natural and interesting question is whether results in \cite{T} remain
valid for $a>0$.
Our answer is affirmative and given for $N=3$. To our best knowledge, this
kind of problems has not been considered before.

 We will use the following notation: 
$S$ is the best Sobolev constant for the embedding from $H_0^1(\Omega )$ to
$L^{6}(\Omega );\|\cdot\|$ is the norm of $H_0^1(\Omega )$ induced by the
product $(u,v)=\int_{\Omega }\nabla u\nabla vdx$;
$\|\cdot \|_{-}$ and
$|\cdot| _{p}:=( \int_{\Omega }|\cdot|^{p}dx) ^{1/p}$
are the norms in $H^{-1}$and $L^{p}(\Omega )$ for $1\leq p<\infty $
respectively; we denote the space $H_0^1(\Omega )$ by $H$ and the
integrable $\int_{\Omega }udx$ by $\int u$; $B_c^{r}$ is the ball of
center $c$ and radius $r$; $o_n( 1) $ denotes any quantity
which tends to zero as $n$ tends to infinity, 
$O( \varepsilon ^{\alpha }) $ means that 
$| O( \varepsilon ^{\alpha })\varepsilon ^{-\alpha }| \leq K$ for some 
constant $K>0$ and $o( \varepsilon ^{\alpha }) $ means 
$| o( \varepsilon ^{\alpha }) \varepsilon ^{-\alpha }| \to0$ as $\epsilon \to 0$. 

In what follows, we fix $b>0$ and consider $a$ as a positive parameter.
To state our main results, we need the following hypothesis
\begin{itemize}
\item[(H1)] 
$| \int fv| <K_a(v)$, for all $v\in H$  such that $|v|_6=1$,
where
\[
K_a(v):=10^{-5/2}[12a^2\|v\|^{8}+80b\|v\|^2
+4a\|v\|^4A_a(v)][3a\|v\|^4+A_a(v)]^{1/2}
\]
with $A_a(v):=\|v\|( 9a^2\|v\|^{6}+20b) ^{1/2}$.
\end{itemize}
We shall prove the following results.

\begin{theorem} \label{thm1}
Assume that $f\neq 0$ satisfies {\rm (H1)}. Then problem 
\eqref{eP} admits at least one weak solution in $H$.
 It is nonnegative if $f$ is also nonnegative.
\end{theorem}

\begin{theorem} \label{thm2}
Under hypothesis of Theorem \ref{thm1} and for $a$ a small positive number,  
problem \eqref{eP} admits at least two weak solutions in $H$. They are
nonnegative if $f$ is also nonnegative.
\end{theorem}

\begin{remark} \label{rmk1} \rm 
In dimension 1 and 2, our problem becomes subcritical and standard
compactness argument applies to get the existence of solutions. 
This also happens for $f\equiv 0$.
For dimensions higher than three, the problem under consideration turns out
to be ``supercritical" thus no existence result is  suspected directly via
variational methods.

Theorem \ref{thm1} remains valid when $f$ satisfies
\[
\big| \int fv\big| \leq K_a(v),\text{ for all }v\in H\quad\text{such that }|v|_6=1.
\]
\end{remark}

These remarks clarify the purpose of restricting this study to dimension three 
in this paper.
This work is organized as follows: 
in Section 2 we give the definition of
Palais-Smale condition and some preliminary results which we will use later.
Section 3 is devoted to the proofs of Theorems \ref{thm1} and \ref{thm2}.


\section{Preliminary results}

We define the energy functional corresponding to  problem \eqref{eP}
by
\[
I_a(u)=\frac{1}{2}\widehat{M}( \| u\| ^2) -
\frac{1}{6}|u|_6^{6}-\int fu,\quad \text{for all }u\in H
\]
where $\widehat{M}(t)$ is the primitive of $M(t)=at+b$  with
$\widehat{M}(0)=0$.
 It is clear that $I_a$ is well defined and of $C^1$ on $H$ and its
critical points are weak solutions of problem \eqref{eP} i.e. they
satisfy:
\[
( a\|u\|^2+b) \int \nabla u\nabla v-\int |
u| ^4uv-\int fv=0,\text{ for all }v\in H.
\]
The functional $I_a$ is not bounded from below on $H$ but it is on a
subset of $H$. A good candidate for an appropriate subset of $H$ is the so
called Nehari manifold defined by
\[
\mathcal{N}=\{ u\in H\backslash \{0\}:\langle I_a'(u),
 u\rangle =0\} .
\]
Let $h_u(t)=I_a(tu)$ for $t\in\mathbb{R}^{\ast }$ and $u\in H\backslash \{0\}$.
These maps are known as fibering
maps and were first introduced by Dr\'{a}bek and Pohozaev \cite{DP}.
The set $\mathcal{N}$ is closely linked to the behavior of $h_u(t)$, for more
details see for example \cite{CKW}.

It is natural to split $\mathcal{N}$ into three subsets:
\begin{gather*}
\mathcal{N}^{+}:=\{ u\in \mathcal{N}: h_u''(1)>0\} ,\quad
\mathcal{N}^{0}:=\{ u\in \mathcal{N}:h_u''(1)=0\}, \\
\mathcal{N}^{-}:=\{ u\in \mathcal{N}: h_u''(1)<0\} ,
\end{gather*}
where $h_u'' (t)=-5|u|_6^{6}t^4+3a\|u\|^4t^2+b\|u\|^2$.
These subsets correspond to local minima, points of inflexion and local maxima 
of $I_a$ respectively.

\begin{definition} \label{def1} \rm
A sequence $(u_n)$ is said to be a Palais-Smale sequence at level $c$
((P-S)$_c$ in short) for $I$ in $H$ if
\[
I(u_n)=c+o_n( 1) \text{ and }I'(u_n)=o_n(1) \text{ in }H^{-1}.
\]
We say that $I$ satisfies the Palais-Smale condition at level $c$ if any
(P-S)$_c$ sequence for $I$ has a convergent subsequence in $H$.
\end{definition}

Put 
\[
H_u(t)=h_u'(t)+\int fu=-|u|_6^{6}t^{5}+a\|u\|^4t^3+b\|u\|^2t.
\]
The function $H_u(t)$ attains its maximum $\widetilde{K}_a(u)$ at the
point $t_{a,{\rm max} }^{u}$ where
\[
\widetilde{K}_a(u)
:=10^{-5/2}|u|_6^{-9}[12a^2\|u\|^{8}+80b|u|_6^{6}\|u\|^2+4a\|u\|^4
\widetilde{A}_a(u)][3a\|u\|^4+\widetilde{A}_a(u)]^{1/2}
\]
and
\[
t_{a,{\rm max} }^{u}=10^{-1/2}|u|_6^{-3}( 3a\|u\|^4+\widetilde{A}
_a(u)) ^{1/4}
\]
with $\widetilde{A}_a(u):=\|u\|(9a^2\|u\|^{6}+20b|u|_6^{6}) ^{1/2}$.

For $a\geq 0$, let
\[
\widetilde{\mu }_{a,f}:=\inf_{v\in H\backslash\{ 0\}}
\{ \widetilde{K_a}(v)-| \int fv|\}, \quad
\mu _{a,f}:=\inf_{|v|_6=1} \{ K_a(v)-\int fv\}.
\]

\begin{remark} \label{rmk2} \rm
(i) If $\widetilde{\mu }_{a,f}>0$ then $\mu _{a,f}>0$.

(ii) We have, for $a>0$, $\widetilde{\mu }_{a,f}\geq \widetilde{\mu }_{0,f}$.
Under the hypothese (H1) with $a=0$, Tarantello has proved that 
$\mu _{0,f}>0$. Thus we deduce that $\widetilde{\mu }_{a,f}>0$.
\end{remark}

The following lemmas play crucial roles in the sequel.

\begin{lemma} \label{lem1}
Suppose {\rm (H1)} holds. Then, for any $u\in H\backslash
\{0\}$, there exist three unique values $t_1^{+}=t_1^{+}(u)$,
$t^{-}=t^{-}(u)\neq 0$ and $t_2^{+}=t_2^{+}(u)$ such that:
\begin{itemize}
\item[(i)] $t_1^{+}<-t_{a,{\rm max} }^{u}$,
$t_1^{+}u\in \mathcal{N}^{-}$, and 
$I_a(t_1^{+}u)=\underset{t\leq -t_{a,{\rm max} }^{u}}{\max }I_a(tu)$,

\item[(ii)] $-t_{a,{\rm max} }^{u}<$ $t^{-}<t_{a,{\rm max} }^{u}$, 
$t^{-}u\in \mathcal{N}^{+}$
and $I_a(t^{-}u)=\min\limits_{| t| \leq t_{a,{\rm max} }^{u}} I_a(tu)$

\item[(iii)] $t_2^{+}>t_{a,{\rm max} }^{u},t_2^{+}u\in \mathcal{N}^{-}$ and $
I_a(t_2^{+}u)=\max\limits_{t\geq t_{a,{\rm max} }^{u}} I_a(tu)$.
\end{itemize}
\end{lemma}

\begin{proof}
An easy computation shows that $H_u(t)$ is concave for $t>0$ and attains its
maximum $\widetilde{K}_a(u)$ at $t_{a,{\rm max} }^{u}$. As $H_u(t)$ is odd
and under the hypothesis (H1) we obtain the desired results.
\end{proof}

For $t>0$, we have
\[
\Psi (tu)=t\Psi (u),\quad \text{where }\Psi (u)=\widetilde{K_a}(u)-|
\int fu| ,
\]
and for a given $\gamma >0$, we derive that
\begin{equation} \label{e2.1}
\inf_{|u|_6\geq \gamma } \Psi (u)\geq \gamma \widetilde{\mu }_{a,f}.
\end{equation}
In particular if $f$ satisfies (H1) this infimum is bounded away
from zero.

\begin{lemma} \label{lem2}
If $f$ satisfies {\rm (H1)}, then $\mathcal{N}^{0}=\emptyset $.
\end{lemma}

\begin{proof}
Arguing by contradiction we assume that there exists $u\in \mathcal{N}^{0}$,
i.e.,
\begin{equation} \label{e2.2}
3a\| u\| ^4+b\| u\| ^2=5|
u| _6^{6};
\end{equation}
thus, we obtain:
\[
\widetilde{A}_a(u)=3a\| u\| ^4+2b\|
u\| ^2,\text{ and }( t_{a,{\rm max} }^{u}) ^2=1\,.
\]
Consequently,
\begin{equation} \label{e2.3}
\Psi (u)=\widetilde{K_a}(u)-| \int fu| \leq \widetilde{
K_a}(u)-\int fu=H_u(1)-\int fu=h_u'(1)=0\,.
\end{equation}
Condition \eqref{e2.2} implies that
\[
| u| _6\geq ( \frac{b}{5}S) ^{1/4}:=\gamma .
\]
From \eqref{e2.1} and \eqref{e2.3} we obtain
\[
0<\gamma \widetilde{\mu }_{a,f}\leq \Psi (u)=0,
\]
which yields a contradiction.
\end{proof}

\begin{lemma} \label{lem3}
Suppose that $f\neq 0$ satisfies {\rm (H1)}, then for each $u\in
\mathcal{N}$, there exist $\varepsilon >0$ and a differentiable function 
$t:B(0,\varepsilon )\subset H\to\mathbb{R}^{+}$ such that $t(0)=1$, 
$t(v)(u-v)\in \mathcal{N}$ for $\|v\| <\epsilon $ and
\begin{equation} \label{e2.4}
\langle t'(0),v\rangle =\frac{2( 2a\|
u\| ^2+b) \int \nabla u\nabla v-6b\int |
u| ^4uv-\int fv}{3a\|u\|^4+b\|u\|^2-5|u|_6^{6}}.
\end{equation}
\end{lemma}

\begin{proof}
Define the map $F:\mathbb{R}\times H\to\mathbb{R}$, by
\[
F(s,w)=as^3\|u-w\|^4+bs\|u-w\|^2-s^{5}|u-w|_6^{6}-\int f(u-w).
\]
Since $F(1,0)=0$, $\frac{\partial F}{\partial s}
(1,0)=3a\|u\|^4+b\|u\|^2-5|u|_6^{6}\neq 0$ and applying the implicit
function theorem at the point $(1,0)$, we get the desired result.
\end{proof}

Define
\begin{equation} \label{e2.5}
c_0=\underset{v\in \mathcal{N}^{+}}{\inf }I_a( v), \quad 
c_1=\underset{v\in \mathcal{N}^{-}}{\inf }I_a( v) .
\end{equation}
Moreover if $u_0$ is a local minimum for $I_a$ then we have 
$3a\| u_0\| ^4+b\| u_0\|^2-5| u_0| _6^{6}\geq 0$ and since
$\mathcal{N}^{0}=\emptyset $, we obtain $u_0\in \mathcal{N}^{+}$.
Consequently $c_0=\underset{u\in \mathcal{N}}{\inf }I_a( u) $.

\begin{lemma} \label{lem4}
The functional $I_a$ is coercive and bounded from below on $\mathcal{N}$.
\end{lemma}

\begin{proof}
For $u\in \mathcal{N}$, we have $a\| u\| ^4+b\|u\| ^2=| u| _6^{6}+\int fu$.
Therefore, we get
\begin{align*}
I_a(u)
&=\frac{a}{12}\| u\| ^4+\frac{b}{3}\| u\| ^2-\frac{5}{6}\int fu \\
&\geq \frac{b}{3}\| u\| ^2-\frac{5}{6}\|f\| _{-}\| u\| , \\
&\geq \frac{-25}{48b}\| f\| _{-}^2,
\end{align*}
Thus $I_a$ is coercive and bounded from below on $\mathcal{N}$.
\end{proof}

In particular, we have $c_0\geq $ $\frac{-25}{48b}\| f\|_{-}^2$.
To prove that $c_0<0$, we need an upper bound for $c_0$. For
this, consider $v\in H$ the unique solution of the equation $-\Delta u=f$.
Then for $f\not\equiv 0$ we have 
$\int fv=\| v\| ^2=\| f\| _{-}^2$.

Let $t_0=t^{-}(v)$, $v\in H\backslash \{0\}$ defined as in Lemma \ref{lem1}. So 
$t_0v\in \mathcal{N}^{+}$ and consequently we have
\begin{align*}
I_a(t_0v) 
&= -\frac{3a}{4}t_0^4\| v\| ^4-\frac{b}{2}t_0^2\| v\| ^2+\frac{5}{6}t_0^{6}|
v| _6^{6} \\
&\leq -\frac{a}{4}t_0^4\| v\| ^4-\frac{b}{3}
t_0^2\| v\| ^2<0,
\end{align*}
thus $c_0<0$.

\begin{lemma} \label{lem5}
Let $f$ verifying {\rm (H1)}, then there exist minimizing sequences 
$(u_n)\subset \mathcal{N}^{+}$ and $(v_n)\subset \mathcal{N}^{-}$ such
that
\begin{itemize}
\item[(i)] $I_a( u_n) <c_0+\frac{1}{n}$ and $I_a( w)
\geq I_a( u_n) -\frac{1}{n}\| w-u_n\| $
for all $w\in \mathcal{N}^{+}$.

\item[(ii)] $I_a( v_n) <c_1+\frac{1}{n}$ and $I_a( w)
\geq I_a( v_n) -\frac{1}{n}\| w-v_n\| $
for all $w\in \mathcal{N}^{-}$.
\end{itemize}
\end{lemma}

\begin{proof}
It is easy to prove that $I_a$ is bounded in $\mathcal{N}$, then by using
the Ekeland Variational Principle to minimization problems \eqref{e2.5}, we get
minimizing sequences $(u_n)\subset \mathcal{N}^{+}$ and $(v_n)\subset
\mathcal{N}^{-}$ satisfying (i) and (ii) respectively.
\end{proof}

Let $(u_n)\subset \mathcal{N}^{+}$ be the minimizing sequence obtained in
the above lemma. For $n$ large enough, we have
\[
I_a( u_n) =\frac{a}{12}\| u_n\| ^4+\frac{
b}{3}\| u_n\| ^2-\frac{5}{6}\int fu_n<c_0+\frac{1}{n
}\leq -\frac{b}{3}t_0^2\| f\| _{-}^2,
\]
this implies
\begin{equation} \label{e2.6}
\int fu_n\geq \frac{2}{5}bt_0^2\| f\| _{-}^2>0,
\end{equation}
and consequently we have
\begin{equation} \label{e2.7}
\frac{2}{5}bt_0^2\| f\| _{-}
\leq \| u_n\| \leq \frac{5}{2b}\| f\| _{-}.
\end{equation}
So, we deduce that $( u_n) $ is bounded in $H$.

\begin{lemma} \label{lem6}
Let $f$ verifying {\rm (H1)}, then $\| I_a'(u_n) \| $ tends to $0$ as 
$n$ tends to $+\infty $.
\end{lemma}

\begin{proof}
Assume that $\| I_a'( u_n) \| >0$
for $n$ large, by applying Lemma \ref{lem3} with $u=u_n$ and
 $w=\delta \frac{I_a'( u_n) }{\| I_a'(u_n) \| }$, 
$\delta >0$ small, we find $t_n(\delta ):=t
[ \delta \frac{I_a'( u_n) }{\|
I_a'( u_n) \| }] $, such that
\[
w_{\delta }=t_n(\delta )\big[ u_n-\delta \frac{I_a'(
u_n) }{\| I_a'( u_n) \| }
\big] \in \mathcal{N}.
\]
From the Ekeland Variational Principle, we have
\begin{align*}
\frac{1}{n}\| w_{\delta }-u_n\|
&\geq I_a( u_n) -I_a( w_{\delta }) \\
&=(1-t_n(\delta ))\langle I_a(w_{\delta }),u_n\rangle
 +\delta t_n(\delta )\langle I_a'(w_{\delta }),\frac{I_a'(u_n) }{\| I_a'( u_n) \| }
\rangle +o_n( \delta ) .
\end{align*}
Dividing by $\delta $ and passing to the limit as $\delta $ goes to zero, we
get
\[
\frac{1}{n}(1+| t_n'(0)| \|
u_n\| )\geq -t_n'(0)\langle I_a'(u_n),\text{ }u_n\rangle +\| I_a'(
u_n) \| =\| I_a'( u_n)\| ,
\]
where $t_n'(0)=\langle t'(0),\frac{I_a'( u_n) }{\| I_a'( u_n)
\| }\text{ }\rangle $. Thus from \eqref{e2.7}, we conclude that
\[
\| I_a'( u_n) \| \leq \frac{C}{n} ( 1+| t_n'(0)| ) .
\]
We claim that $| t_n'(0)| $ is bounded
uniformly on $n$; indeed, since $(u_n)$ is a bounded sequence,
from \eqref{e2.4} and the estimate \eqref{e2.7}, we have
\[
| t_n'(0)| \leq \frac{C}{|3a\| u_n\| ^4+b\| u_n\|
^2-5| u_n| _6^{6}| }.
\]
Hence we must prove that
$| 3a\| u_n\|^4+b\| u_n\| ^2-5| u_n|_6^{6}| $ is bounded away from zero.
 Arguing by contradiction,
assume that for a subsequence still called $(u_n)$, we have
\begin{equation} \label{e2.8}
3a\| u_n\| ^4+b\| u_n\|^2-5| u_n| _6^{6}=o_n(1).
\end{equation}
From \eqref{e2.7} and \eqref{e2.8} we derive that
\[
| u_n| _6\geq \gamma ,\text{ for a suitable constant }
\gamma
\]
In addition \eqref{e2.8} and the fact that $u_n\in \mathcal{N}$ also give
\[
\int fu_n=-2a\| u_n\| ^4+4| u_n|_6^{6}+o_n(1),
\]
which together with the definition of $\widetilde{\mu }_{a,f}$ imply that
\begin{align*}
0 &<\gamma \widetilde{\mu }_{a,f}\leq \gamma (\widetilde{K_a}(u_n)-\int
fu_n)+o_n(1) \\
&=\gamma h_{u_n}'(1)+o_n(1)
=o_n(1).
\end{align*}
which is absurd. Thus $\| I_a'( u_n)
\| $ tends to $0$ as $n$ tends to $\infty $.
\end{proof}

\section{Proofs of Theorems \ref{thm1} and \ref{thm2}}

\subsection{Existence of a local minimizer on $\mathcal{N}^{+}$}

In this subsection, we prove that $I_a$ achieves a local minimum in
 $\mathcal{N}^{+}$ by the Ekeland Variational Principle.

\begin{proof}[Proof of Theorem \ref{thm1}]
Since $( u_n) $ is bounded in $H$, passing to a subsequence if
necessary, we have $u_n\rightharpoonup $ $u_0$ weakly in $H$, then we
get $\langle I_a'( u_0) ,w\rangle =0$,
for all $w\in H$. So $u_0$ is a weak solution for 
\eqref{eP}.

From \eqref{e2.6}, we deduce that $\int fu_0>0$, then 
$u_0\in H\backslash \{ 0\} $ and in particular $u_0\in \mathcal{N}$.
Thus
\[
c_0\leq I_a( u_0) =\frac{a}{12}\| u_0\|
^4+\frac{b}{3}\| u_0\| ^2-\frac{5}{6}\int fu_0\leq
\liminf_{n\to \infty } I_a(u_n)=c_0,
\]
then $c_0=I_a( u_0) $. It follows that $( u_n)$ converges strongly to
 $u_0$ in $H$ and necessarily $u_0\in \mathcal{N}^{+}$.
To conclude that $u_0$ is a local minimum of $I_a$, let us recall
that for every $u\in H$, we have
\[
I_a(su)\geq I_a( t^{-}u) \quad \text{for every }0<s<t_{a,{\rm max}}^{u},
\]
in particular for $u=u_0\in \mathcal{N}^{+}$, we have
 $t^{-}=1<t_{a,{\rm max} }^{u_0}$. Choose $\varepsilon >0$ sufficiently
small to have $1<t_{a,{\rm max}}^{u_0-w}$ and $t(w)$ satisfying
$t(w)( u_0-w) \in \mathcal{N} $ for every $\| w\| <\varepsilon $.
Since $t(w)\to 1 $ as $\| w\| \to 0$, we can always assume that
\[
t(w)<t_{a,{\rm max} }^{u_0-w}\quad \text{for every $w$ such that }\|
w\| <\varepsilon ,
\]
so $t(w)( u_0-w) \in \mathcal{N}^{+}$ and for $0<s<t_{a,{\rm max}}^{u_0-w}$,
 we have
\[
I_a( s( u_0-w) ) \geq I_a( t(w)(u_0-w) ) \geq I_a( u_0) ,
\]
taking $s=1$, we conclude that $I_a( u_0-w) \geq I_a(u_0)$,
 for all $w\in H$ such that
$\| w\|<\varepsilon$.
\end{proof}

To see that $u_0\geq 0$ when $f\geq 0$, it suffices to take 
$t_0=t^{-}(| u_0| )$ such that $t_0| u_0| \in \mathcal{N}^{+}$. 
This implies that necessarily
\[
I_a(t_0| u_0| )\leq I_a(|u_0| )\leq I_a(u_0).
\]
Consequently, we can always take $u_0\geq 0$.

\subsection{Existence of a local minimizer on $\mathcal{N}^{-}$}

This subsection is devoted to the existence of a second solution $u_1$ in 
$\mathcal{N}^{-}$ via Mountain Pass Lemma such that 
$c_1=I_a(u_1) $. First we determine the good level for covering the
Palais-Smale condition.

The best Sobolev constant $S$ is attained in $\mathbb{R}^3$ by
\[
U_{\varepsilon ,x_0}(x)=\varepsilon ^{1/2}( \varepsilon
^2+| x-x_0| ^2) ^{-1/2},
\]
where $x_0\in \Omega $ and $\varepsilon >0$.
We have the following important result.

\begin{lemma} \label{lem7}
Let $f$ satisfying {\rm (H1)}, then $I_a$ satisfies the (P-S)$_c$
condition for
\[
c<c^{\ast }=\frac{ab}{4}S^3+\frac{a^3}{24}S^{6}+\frac{b}{6}SE_1+\frac{
a^2}{24}S^4E_1+c_0,
\]
where $E_1=( a^2S^4+4bS) ^{1/2}$.
\end{lemma}

\begin{proof}
Let $( u_n) $ be a (P-S)$_c$ sequence with $c<c^{\ast }$,
then $( u_n) $ is a bounded sequence in $H$. Thus it has a
subsequence still denoted $( u_n) $ such that 
$u_n\rightharpoonup u$ in $H$, $u_n\to u$ strongly in 
$L^{s}( \Omega ) $ for all $1\leq s<6$ and $u_n\to u$
a.e. in $\Omega $.

Let $w_n=u_n-u$. From the Brezis-Lieb Lemma \cite{BL}, one has:
\begin{gather*}
\| u_n\| ^2=\| w_n\| ^2+\|u\| ^2+o_n(1),\quad
\| u_n\|^4=\| w_n\| ^4+2\| w_n\| ^2\| u\| ^2+\| u\| ^4+o_n(1),\\
| u_n| _6^{6}=| w_n|_6^{6}+| u| _6^{6}+o_n(1).
\end{gather*}
Since $I_a(u_n)=c+o_n(1)$, we get
\[
\frac{a}{4}\| w_n\| ^4+\frac{b}{2}\|
w_n\| ^2+\frac{a}{2}\| w_n\| ^2\|
u\| ^2-\frac{1}{6}| w_n|
_6^{6}=I_a(u_n)-I_a(u)=c-I_a(u)+o_n(1)\,.
\]
By the fact that $I_a'(u_n)=o_n(1)$ and
$\langle I_a'(u),u\rangle =0$, we obtain
\[
a\| w_n\| ^4+b\| w_n\|^2+2a\| w_n\| ^2\| u\|^2-| w_n| _6^{6}=o_n(1)\,.
\]
Assume that $\| w_n\| \to l$ with $l>0$, it
follows that
\[
| w_n| _6^{6}=al^4+bl^2+2al^2\|u\| ^2+o_n(1)\,.
\]
From the definition of $S$, we have
\[
\| w_n\| ^2\geq S| w_n| _6^2,\quad \text{for all }n\,..
\]
As $n\to +\infty $, we deduce that
\[
l^2\geq \frac{a}{2}S^3+\frac{1}{2}S\big( a^2S^4+4S(b+2a\|u\| ^2)\big) ^{1/2}.
\]
Consequently we obtain
\begin{align*}
c &=\frac{a}{12}l^4+\frac{b}{3}l^2+\frac{a}{6}l^2\|
u\| ^2+I_a(u) \\
&\geq \frac{a}{12}l^4+\frac{b}{3}l^2+c_0 \\
&\geq \frac{ab}{4}S^3+\frac{a^3}{24}S^{6}+\frac{b}{6}SE_1+\frac{a^2
}{24}S^4E_1+c_0=c^{\ast }
\end{align*}
which is a contradiction. Therefore $l=0$, then $u_n\to u$
strongly in $H$.
\end{proof}

Now, we shall give some useful estimates of the extremal functions. 
Let $ \phi \in C_0^{\infty }( \Omega ) $ such that $\phi (x)=1$ for
 $x\in B_{x_0}^{r}$, $\phi (x)=0$ for
 $x\in\mathbb{R}^3\backslash B_{x_0}^{2r}$, $0\leq \phi \leq 1$ and 
$| \nabla\phi | \leq C$.

Set $u_{\varepsilon ,x_0}( x) =\phi (x)U_{\varepsilon,x_0}(x)$.
The following estimates are obtained in \cite{BN}, as $\varepsilon $ tends
to $0$:
\[
| u_{\varepsilon ,x_0}| _6^{6}=A+O( \varepsilon
^3) \quad\text{and}\quad
\| u_{\varepsilon ,x_0}\|^2=B+O( \varepsilon ) ,
\]
where
\[
A=\int_{\mathbb{R}^3}( 1+| x-x_0| ^2) ^{-3}, \quad
B=\int_{\mathbb{R}^3}| \nabla U_{1,x_0}( x) | ^2,
\]
and from \cite{T}, we have
 $\int u_{\varepsilon ,x_0}^{5}u_0=O(\varepsilon ^{1/2}) +o( \varepsilon ^{1/2}) $.

In the search of our second solution, it is natural to show that 
$c_1<c^{\ast }$. For this let $\Omega '\subset \Omega $ a be set of
positive measure such that $u_0>0$ on $\Omega '$ (if not replace 
$u_0$ and $f$ by $-u_0$ and $-f$ respectively), where $u_0$ is given in
Theorem \ref{thm1}.

\begin{lemma} \label{lem8}
Assume that the hypothesis {\rm (H1)} is satisfied, then there exist 
$a_0$ and $\varepsilon _0$ small enough such that for every 
$0<\varepsilon <\varepsilon _0$ and $0<a<a_0$ we have 
$I_a(u_0+tu_{\varepsilon ,x_0}) <c^{\ast }$ for all $t>0$.
\end{lemma}

\begin{proof}
From the above estimates and the Holder Inequality, we obtain
\begin{align*}
&I_a( u_0+tu_{\varepsilon ,x_0}) \\
&= I_a( u_0)+\frac{a}{4}t^4\| u_{\varepsilon ,x_0}\| ^4+\frac{b}{2
}t^2\| u_{\varepsilon ,x_0}\| ^2-\frac{1}{6}
t^{6}| u_{\varepsilon ,x_0}| _6^{6}-\frac{t^{5}}{6}
\int u_{\varepsilon }^{5}u_0\\
&\quad + at^2\Big[ \Big( \int \nabla u_0\nabla u_{\varepsilon
}\Big) ^2+\| u_{\varepsilon }\| ^2\Big(\frac{1}{2}
\| u_0\| ^2+t\int \nabla u_0\nabla
u_{\varepsilon }\Big)\Big] +o( \varepsilon ^{1/2}) \\
&\leq I_a( u_0) +\frac{a}{4}t^4B^2+\frac{b}{2}t^2B-
\frac{1}{6}t^{6}A-\frac{t^{5}}{6}O( \varepsilon ^{1/2}) + \\
&\quad +at^2\big[ \frac{3}{2}\| u_0\|
^2B+tB^{3/2}\| u_0\| \big] +o( \varepsilon
^{1/2}) \\
&= c_0+Q_{\varepsilon}( t) +R(t),
\end{align*}
where
\[
Q_{\varepsilon}( t) =-\frac{1}{6}At^{6}+\frac{a}{4}B^2t^4+
\frac{b}{2}Bt^2-\frac{t^{5}}{6}O( \varepsilon ^{1/2}) +o(
\varepsilon ^{1/2}) ,
\]
and
\[
R(t)=a\big[ \frac{3}{2}t^2\| u_0\|
^2B+t^3B^{3/2}\| u_0\| \big] .
\]
We know that $\lim_{t\to +\infty } Q_{\varepsilon}( t) =-\infty $,
 and $Q_{\varepsilon}( t) >0$ for $t$ near $0$, so
$\sup_{t\geq 0} Q_{\varepsilon}( t) $ is achieved for
$t=T_{\varepsilon }>0$ and $ T_{\varepsilon }$ satisfies:
\[
-AT_{\varepsilon }^{5}+aB^2T_{\varepsilon }^3+bBT_{\varepsilon }=O(
\varepsilon ^{1/2}) .
\]
Also $Q_{_0}( t) $ attains its maximum at $T_0$ given by
\[
T_0^2=\frac{aB^2+( a^2B^4+4bAB) ^{1/2}}{2A}.
\]
It is clear that $T_{\varepsilon }$ tends to $T_0$ as $\varepsilon $ goes
to $0$. Write $T_{\varepsilon }=T_0(1\pm \delta _{\varepsilon})$,
hence $\delta _{\varepsilon}$ tends to $0$ as $\varepsilon $ goes to $0$.

Moreover, since $I_a( u_0+tu_{\varepsilon }) \to
-\infty $ as $t$ approaches $\infty $, there exists $T_{\varepsilon }<T_1$
such that
\[
I_a( u_0+tu_{\varepsilon ,x_0}) \leq c^{\ast
}+Q_{\varepsilon}( T_{\varepsilon }) +\sup_{t<T_1}R(t).
\]
On the other hand, we have
\begin{equation} \label{e3.1}
\begin{aligned}
Q_{\varepsilon}( T_{\varepsilon })
&= -\frac{1}{6}AT_{\varepsilon }^{6}+\frac{a}{4}B^2T_{\varepsilon }^4+\frac{b}{2}
BT_{\varepsilon }^2-O( \varepsilon ^{1/2}) +o( \varepsilon
^{1/2})   \\
&=-\frac{1}{6}AT_0^{6}+\frac{a}{4}B^2T_0^4+\frac{b}{2}BT_0^2\pm
aT_0^4B^2\delta _{\varepsilon}\pm bT_0^2B\delta
_{\varepsilon}\mp T_0^{6}A\delta _{\varepsilon}\\
&\quad -O(\varepsilon ^{1/2}) +o( \varepsilon ^{1/2})   \\
&=-\frac{1}{6}AT_0^{6}+\frac{a}{4}B^2T_0^4+\frac{b}{2}
BT_0^2-O( \varepsilon ^{1/2}) +o( \varepsilon
^{1/2}) .
\end{aligned}
\end{equation}
Now substituting the expression of $T_0$ in \eqref{e3.1}, we obtain
\begin{align*}
Q_{\varepsilon}( T_{\varepsilon })
&= \frac{abB^3}{4A}+
\frac{b(a^2B^{6}+4bB^3A)^{1/2}}{6A}+\frac{a^3B^{6}}{24A^2}+\frac{
a^2(a^2B^{12}+4bB^{9}A)^{1/2}}{24A^2}\\
&\quad -O( \varepsilon^{1/2}) +o( \varepsilon ^{1/2}) \\
&= \frac{ab}{4}S^3+\frac{a^3}{24}S^{6}+(\frac{b}{6}S+\frac{a^2}{24}
S^4)(a^2S^4+4bS)^{1/2}-O( \varepsilon ^{1/2}) +o(
\varepsilon ^{1/2}) \\
&\leq  c^{\ast }-c_0-O( \varepsilon ^{1/2}) +o(
\varepsilon ^{1/2}) .
\end{align*}
Thus we have
\begin{align*}
I_a( u_0+tu_{\varepsilon ,x_0})
&\leq c^{\ast }-O(
\varepsilon ^{1/2}) +o( \varepsilon ^{1/2})
+\sup_{t<T_1}R(t) \\
&\leq c^{\ast }-O( \varepsilon ^{1/2}) +o( \varepsilon
^{1/2}) +aK,
\end{align*}
where $K:=\frac{3}{2}T_1^2\| u_0\|^2B+T_1^3B^{3/2}\| u_0\| $.

Consequently, there exist $a_0$ and $\varepsilon _0$ small enough such
that
$I_a( u_0+tu_{\varepsilon ,x_0}) <c^{\ast }$ for every
$0<\varepsilon <\varepsilon _0$ and $0<a<a_0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
By Lemma \ref{lem1}, there exists an unique $t^{+}( u) >0$ such that 
$ t^{+}( u) u\in \mathcal{N}^{-}$ and 
$I_a(t^{+}u)\geq I_a(tu)$, for all 
$| t| \geq t_{a,{\rm max} }^{u}$ and every
$u\in H$ such that $\| u\| =1$.

The extremal property of $t^{+}( u) $ and its uniqueness give
that it is a continuous function of $u$.
Set
\[
V_1=\{ 0\} \cup \{ u:\|u\| <t^{+}( \frac{u}{\| u\| }) \} ,\quad
V_2=\{ u:\| u\|>t^{+}( \frac{u}{\| u\| }) \} .
\]
As in \cite{T}, we remark that under the condition (H1),
we have $H\backslash \mathcal{N}^{-}=V_1\cup V_2$ and
$\mathcal{N}^{+}\subset V_1$, $u_0\in V_1$ and
$u_0+t_0u_{\varepsilon }\in V_2$ for a $t_0>0$, carefully chosen.

Let $\Gamma =\{ h:[ 0,1] \to H\text{ continuous}, 
h(0)=u_0, h(1)=u_0+t_0u_{\varepsilon }\} $.
It is obvious that $h:[ 0,1] \to H$  given by 
$h(t)=u_0+tt_0u_{\varepsilon }$ belongs to $\Gamma $.
 We conclude that
\[
c=\underset{h\in \Gamma }{\inf }\underset{t\in [ 0,1] }{\max }
I_a(h(t))<c^{\ast }.
\]
As the range of any $h\in $ $\Gamma $ intersects $\mathcal{N}^{-}$, one has
$c\geq c_1$.

Applying again the Ekeland Variational Principle, we obtain a minimizing
sequence ($u_n)\subset \mathcal{N}^{-}$ such that
\[
I_a(u_n)\to c_1\quad \text{and}\quad
\| I_a'(u_n) \| \to 0.
\]
We also deduce that $c_1$ $<c^{\ast }$.
Consequently, we get a subsequence ($u_{n_{k}})$ of $(u_n)$ and
$u_1\in H $ such that
\[
u_{n_{k}}\to u_1\quad \text{strongly in }H.
\]
This implies that $u_1$ is a critical point for $I_a$, $u_1\in
\mathcal{N}^{-}$ and $I_a(u_1)=c_1$.
\end{proof}

Finally for $f\geq 0$, let $t^{+}=t^{+}(| u_1| )>0$
satisfying $t^{+}| u_1| \in \mathcal{N}^{-}$.
From Lemma \ref{lem1} we have 
$I_a(u_1)=\max\limits_{t\geq t_{a,{\rm max} }}I_a(tu_1)\geq I_a(t^{+}u_1)\geq I_a(t^{+}|
u_1| )$. So we conclude that $u_1\geq 0$.

\subsection*{Acknowledgments}
The authors want to thank an anonymous referee for the careful
reading, which greatly improved this article.


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\end{document}