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\date{}
\title{Nehari Manifold and Existence of Positive Solutions for a
Class of p-Laplacian Problems}
\author{ {Khaled Ben Ali}$^1$ and {Abdeljabbar Ghanmi}$^2$ \\
\small $^1$ Facult\'e des Sciences de Gab\`es  Cit\'e Erriadh, 6072 Gab\`es, Tunisia.\\
\small $^2$ D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis Campus Universitaire, 2092 Tunis, Tunisia.\\
\small E-mail addresses : {\tt benali.khaled@yahoo.fr and Abdeljabbar.ghanmi@lamsin.rnu.tn}}
\begin{document}
\baselineskip16pt \maketitle
\noindent{\small{ Abstract}.\\
This article shows the existence and multiplicity of positive solutions of the $p$-Laplacien problem
$$\displaystyle  -\Delta_{p} u=\frac{1}{p^{\ast}}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u \quad \mbox{for } x\in\Omega;\quad \quad u=0,\quad   \mbox{for } x\in\partial\Omega$$
where $\Omega$ is a bounded open set in $\mathbb{R}^n$ with smooth boundary, $1<q<p<n$, $p^{\ast}=\frac{np}{n-p}$, $\lambda \in \mathbb{R}\backslash \{0\}$ and  $a$ is a smooth function which may change sign in $\overline{\Omega}$. The method is based on
Nehari results on three sub-manifolds of the space $W_{0}^{1,p}$.\\\\
\textbf{Keywords}: Multiple positive solutions; sign-changing weight function; Nehari manifold.\\
\textbf{Mathematics Subject Classification:} 35J35, 35J50, 35J60.\\
\section{Introduction}
In this paper we study the multiplicity of positive solutions of the elliptic equation
$$\textbf{(1)}\;\;\;\left\{
 \begin{array}{ll}
-\Delta_{p} u=\frac{1}{p^{\ast}}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u \;\;\mbox{ in } \; \Omega, \\
u=0\;\;\;\mbox{on}\;\;\;\;\;\;\partial \Omega,
  \end{array}
\right.$$
where $\Omega$ is a bounded domain of $\mathbb{R}^{n}\;(n\geq 3)$, $1<q<p<n$, $p^{\ast}=\frac{np}{n-p}$, $\lambda \in \mathbb{R}\backslash \{0\}$, and the sign changing weight function $a$  satisfies the following condition\\
\textbf{(A)} $a\in C(\Omega)$ with $\|a\|_{\infty}=1$ and  $a^{\pm}:=max(\pm a,0)\not \equiv0$.\\
The study of p-Laplacian equations using the Nehari manifold method sees great increase in number of papers published, we refer to [1-9], and the references therein. More precisely, Wu \cite{R1} concidered the following elliptic equation
$$\textbf{(2)}\left\{
 \begin{array}{ll}
  -\triangle_{p} u=\lambda f(x)|u|^{q-2}u+g(x)|u|^{r-2}u,\;\;\mbox{in}\;\;\Omega, \\
   u=0\;\;\;\;\;\mbox{on}\;\;\;\partial \Omega,
 \end{array}
 \right.$$
where $\Omega$ is a bounded domain of $\mathbb{R}^{n}\;(n\geq 3)$, $1<q<p<r<p^{\ast}\; (p^{\ast}=\frac{np}{n-p}\;\mbox{ if }\;n>p, p^{\ast}=\infty \;\mbox{ if }\;n\leq p)$. The author, prove that there exists $\lambda _{0}>0$ such that for $0<\lambda <\lambda _{0}$, equation (2) has at least two positive solutions.
If the weight functions $f\equiv g\equiv 1$,  the authors Ambrosetti-Brezis-Cerami \cite{R2} have investigate equation (2), they found that there exists $\lambda _{0}>0$ such that equation (2) admit two positive solutions for $\lambda \in (0,\lambda _{0} )$, has a positive solution for $\lambda =\lambda _{0}$ and no positive solution exists for $\lambda>\lambda _{0}$.\\
The starting point on the study of the system (1) is its scalar version
$$\textbf{(3)}\;\;\;\left\{
 \begin{array}{ll}
-\triangle_{p} u=|u|^{p^{\ast}-2}u + \lambda |u|^{q-2}u \;\;\mbox{ in } \; \Omega, \\
u=0\;\;\;\mbox{on}\;\;\;\;\;\;\partial \Omega,
  \end{array}
\right.$$
with $2\leq p\leq q <p^{\ast}$. In a pioneer work Brezis and Nirenberg \cite{R14} schowed that, if $p=q=2$, equation (3) has at least one positive solution provided $n \geq 4$ and $\lambda$ is sufficiently small. In particular, the first multiplicity result for (3) has been achieved by Rey \cite{R14} in the semilinear case. Furthermore, Alves and Ding \cite{R15} obtained the existence of positive
solutions to equation (3) with $ 2\leq  p \leq q <p^{*}$.\\
In this work, motivated by the above works, we give a very simple variational method to prove the existence of at least two nontrivial solutions of problem (1). In fact, we use the decomposition of the Nehari manifold as $\lambda$ vary to prove our main result.\\
Before stating our main result, we need the following assumptions:\\
$\textbf{(H$_{1}$)}$ $F: \overline{\Omega}\times \mathbb{R}\longrightarrow \mathbb{R}$ is a $C^{1}$ function such that
$$F(x,tu) = t^{p^{\ast}} F(x,u) (t > 0)\;\;\mbox{for all}\;\; x\in \overline{\Omega},\;u\in \mathbb{R}.$$
$\textbf{(H$_{2}$)}$ $F(x,0)=\frac{\partial F}{\partial u}(x,0)=0$ and $F^{\pm}(x,u)=max(\pm F(x,u),0)\not \equiv 0$ for all $u\neq 0$.\\
We remark, that Using assumption $\textbf{(H$_{1}$)}$, we have the so-called Euler identity
\begin{equation}\label{1}
u \frac{\partial F(x,u)}{\partial u} =p^{\ast} F(x,u),\;\;\mbox{ and }\;\;|F(x,u)| \leq K |u|^{p^{\ast}}\;\;\mbox{ for some constant}\; K > 0.
\end{equation}
Our main result is the following
\begin{teo}
Under the assumption $\textbf{(A)}$,  $\textbf{(H$_{1}$)}$ and $\textbf{(H$_{2}$)}$, there exists $\lambda_{0}>0$ such that for all $ 0<|\lambda|<\lambda_{0}$, problem (1) has at least two positive solutions.\\
\end{teo}
This paper is organized as follows. In Section 2, we give some notations, preliminaries and we present some technical
lemmas which are crucial in the proof of the Theorem 1. In section 3, we prove our result.

\section{Some notations and preliminaries}
Throughout this paper, we denote by $S_{l}$ the best Sobolev embedding for the
operator $W^{1,p}_{0}(\Omega)\hookrightarrow L^{l} (\Omega)$ is given by
$$S_{l} =\displaystyle\inf_{u\in W^{1,p}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}|\nabla u|^{p}}{\left(\int_{\Omega}|u|^{l}\right)^{\frac{p}{l}}},$$
where $1 < l \leq p^{\ast}$.
In particular, we have
\begin{equation}\label{2}
    \int_{\Omega}|u|^{l}\leq S_{l}^{-\frac{l}{p}}\|u\|^{l}\;\;\mbox{ for all}\;\; u\in W^{1,p}_{0}(\Omega)
\end{equation}
with the standard norm $\displaystyle \| u \|=\left(\int_{\Omega}|\nabla u|^{p}dx\right)^{\frac{1}{p}}.$\\
Problem (1) is posed in the framework of the Sobolev space $E = W^{1,p}_{0}(\Omega)$. Moreover, a function $u$ in $E$ is said to be a weak solution of problem (1) if
$$\int_{\Omega}|\nabla u|^{p-2}\nabla u \nabla \varphi dx -\frac{1}{p^{\ast}}\int_{\Omega}\frac{\partial F(x,u)}{\partial u}\varphi dx -\lambda \int_{\Omega}a|u|^{q-2}u \varphi dx=0, \;\mbox{ for all }\;\varphi\in E.$$
Thus, the corresponding energy functional of problem (1) is defined as $J_{\lambda}:E\to \mathbb{R}$,
$$J_{\lambda}(u)=\frac{1}{p}\|u\|^{p}-\frac{1}{p^{\ast}}\int _{\Omega}F(x,u)dx- \frac{\lambda}{q}\int_{\Omega}a(x)|u|^{q} dx.$$
In order to verify $J_{\lambda}\in C^{1}(E,\mathbb{R})$, we need the following lemmas.
\begin{lemma}Assume that $F\in C^{1}(\overline{\Omega}\times \mathbb{R},\mathbb{R})$ is positively homogenuous of degree $p^{\ast}$, then $\frac{\partial F}{\partial u}\in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ is positively homogenuous of degree $p^{\ast}-1.$
\end{lemma}
\begin{proof}
The proof is the same as that in Chu and Tang \cite{R3}.
\end{proof}
In addition, by  lemma 1, we get the existence of positive constant $M$ such that
\begin{equation}\label{20}
|\frac{\partial F(x,u)}{\partial u}|\leq M|u|^{p^{\ast}-1}.
\end{equation}
 \begin{lemma} (See Proposition 1 in \cite{R5}) Suppose that $\frac{\partial F(x,u)}{\partial u}\in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ verifies condition (\ref{20}). Then, the functional $J_{\lambda}$ belongs to $C^{1}(E,\mathbb{R})$, and
\begin{equation}\label{14}
   \langle J'_{\lambda}(u),u\rangle=\|u\|^{p}-\int_{\Omega}F(x,u)dx-\lambda\int_{\Omega}a(x)|u|^{q}dx.
\end{equation}
\end{lemma}
As the energy functional $J_{\lambda}$ is not bounded below on $E$, it is useful to concider the functional on the Nehari manifold $$N_{\lambda}=\{u\in E\backslash \{0\}: \langle J'_{\lambda}(u),u\rangle=0\},$$
where $ \langle ., . \rangle $ denotes the usual duality between $E$ and $E^\ast=W^{-1,p'}(\Omega)$ (the dual space of the Sobolev space $E$).\\
Thus, $u\in N_{\lambda}$ if and only if
\begin{equation}\label{3}
    \|u\|^{p}-\int_{\Omega}F(x,u)dx-\lambda\int_{\Omega}a(x)|u|^{q}dx=0.
\end{equation}
%Note that $N_{\lambda}$, contains every nonzero solution of problem (1).
Moreover, we have the following result.
\begin{lemma}
The energy functional $J_{\lambda}$ is coercive and bounded below on $N_{\lambda}$.
\end{lemma}
\begin{proof}
If $u\in N_{\lambda}$, then by (\ref{3}) and condition \textbf{(A)} we obtain
\begin{eqnarray*}
  J_{\lambda}(u) &=& \frac{p^{\ast}-p}{p^{\ast}p}\|u\|^{p}-\lambda\frac{p^{\ast}-q}{p^{\ast}q}\int_{\Omega}a(x)|u|^{q}dx \\
   &\geq& \frac{p^{\ast}-p}{p^{\ast}p}\|u\|^{p}-|\lambda|\frac{p^{\ast}-q}{p^{\ast}q}\int_{\Omega}|u|^{q}dx.
\end{eqnarray*}
So, it follows from (\ref{2}) that
\begin{equation}\label{16}
    J_{\lambda}(u)\geq \frac{p^{\ast}-p}{p^{\ast}p}\|u\|^{p}-\frac{|\lambda|}{S_{q}^{\frac{q}{p}}}\frac{p^{\ast}-q}{p^{\ast}q}\|u\|^{q}.
\end{equation}
Thus, $J_{\lambda}$ is coercive and bounded below on $N_{\lambda}$.
\end{proof}
Define $$\phi_{\lambda}(u)=\langle J'_{\lambda}(u),u\rangle.$$
Then, by (\ref{3}) it is easy to see that for $u\in N_{\lambda}$
\begin{eqnarray}
  \langle \phi'_{\lambda}(u),u\rangle &=& p\|u\|^{p}-p^{\ast}\int_{\Omega}F(x,u)dx-\lambda q\int_{\Omega}a(x)|u|^{q}dx \\
  &=& \lambda(p-q)\int_{\Omega}a(x)|u|^{q}dx-(p^{\ast}-p)\int_{\Omega}F(x,u)dx  \label{4}\\
   &=& \lambda(p^{\ast}-q)\int_{\Omega}a(x)|u|^{q}dx-(p^{\ast}-p)\|u\|^{p} \label{5} \\
   &=& (p-q)\|u\|^{p}-(p^{\ast}-q)\int_{\Omega}F(x,u)dx .\label{6}
\end{eqnarray}
Now, we split $N_{\lambda}$ into three parts
$$N^{+}_{\lambda}=\{u\in N_{\lambda}: \langle \phi'_{\lambda}(u),u\rangle >0\},$$
$$N^{0}_{\lambda}=\{u\in N_{\lambda}: \langle \phi'_{\lambda}(u),u\rangle=0\},$$
$$N^{-}_{\lambda}=\{u\in N_{\lambda}: \langle \phi'_{\lambda}(u),u\rangle<0\}.$$
\begin{lemma}
Assume that $u_{0}$ is a local minimizer for  $J_{\lambda}$ on  $N_{\lambda}$ and that  $u_{0}\not \in N^{0}_{\lambda}$. Then,  $J'_{\lambda}(u_{0})=0$ in $E^{*}$.
\end{lemma}
\begin{proof}
 Our proof is almost the same as that in Brown-Zhang [\cite{R8}, Theorem 2.3].
\end{proof}
\begin{lemma}\label{7} We have\\
$(i)$ If $u\in N^{+}_{\lambda}$, then $\lambda\int_{\Omega}a(x)|u|^{q}dx>0.$\\
$(ii)$ If $u\in N^{0}_{\lambda}$, then $\lambda\int_{\Omega}a(x)|u|^{q}dx>0$ and $\int_{\Omega}F(x,u)dx>0.$\\
$(iii)$ If $u\in N^{-}_{\lambda}$, then $\int_{\Omega}F(x,u)dx>0.$
\end{lemma}
\begin{proof}
The proofs are immediate from (\ref{4}), (\ref{5}) and (\ref{6}).
\end{proof}
Let
\begin{equation}\label{17}
\lambda_{0}=\frac {q(p^{\ast}-p)}{p(p^{\ast}-q)} S_{q}^{\frac{q}{p}}\left(\frac{p-q}{K(p^{\ast}-q)}S_{p^{\ast}}^{\frac{p^{\ast}}{p}}\right)^{\frac{p-q}{p^{\ast}-p}},
\end{equation}
then we have the following lemma.
\begin{lemma}
 \label{12} If $0<|\lambda|<\lambda_{0}$, then $N^{0}_{\lambda}=\varnothing.$\\
\end{lemma}
\begin{proof}
Suppose otherwise, that  $0<|\lambda|<\lambda_{0}$  such that $N^{0}_{\lambda}\neq\varnothing$. Then for $u\in N^{0}_{\lambda}$, we have
\begin{eqnarray}
0 \;=\;\langle \phi'_{\lambda}(u),u\rangle &=& \lambda(p^{\ast}-q)\int_{\Omega}a(x)|u|^{q}dx-(p^{\ast}-p)\|u\|^{p} \label{8}\\
&=& (p-q)\|u\|^{p}-(p^{\ast}-q)\int_{\Omega}F(x,u)dx. \label{9}
\end{eqnarray}
From the H\"{o}lder inequality, (\ref{1}) and (\ref{2}), it follows that $$\int_{\Omega}F(x,u)dx\leq \int_{\Omega}|F(x,u)|dx\leq K\int_{\Omega}|u|^{p^{\ast}}dx\leq KS_{p^{\ast}}^{-\frac{p^{\ast}}{p}}\|u\|^{p^{\ast}}.$$
Hence, it follows from (\ref{9}) that
 \begin{eqnarray*}
\|u\|^{p} &=& \frac{p^{\ast}-q}{p-q}\int_{\Omega}F(x,u)dx \\
&\leq& \frac{p^{\ast}-q}{p-q}KS_{p^{\ast}}^{-\frac{p^{\ast}}{p}}\|u\|^{p^{\ast}},
\end{eqnarray*}
then,
\begin{equation}\label{10}
  \|u\|\geq \left(\frac{p-q}{K(p^{\ast}-q)}S_{p^{\ast}}^{\frac{p^{\ast}}{p}}\right)^{\frac{1}{p^{\ast}-p}}.
\end{equation}
On the other hand, from condition (\textbf{A}) ,(\ref{2}) and (\ref{8}) we have
 \begin{eqnarray*}
\|u\|^{p} &=& \lambda\frac{p^{\ast}-q}{p^{\ast}-p}\int_{\Omega}a(x)|u|^{q}dx  \\
&\leq& |\lambda|\frac{p^{\ast}-q}{p^{\ast}-p}KS_{q}^{-\frac{q}{p}}\|u\|^{q},
\end{eqnarray*}
so,
\begin{equation}\label{11}
\|u\|\leq \left(|\lambda|\frac{p^{\ast}-q}{p^{\ast}-p}S_{q}^{-\frac{q}{p}}\right)^{\frac{1}{p-q}}.
\end{equation}
Combining (\ref{10}) and (\ref{11}), we obtain $\lambda_{0}\leq |\lambda|,$ which is a contraduction.
\end{proof}
By lemma \ref{12}, for $0<|\lambda|<\lambda_{0}$, we write $N_{\lambda}=N^{+}_{\lambda}\cup N^{-}_{\lambda}$ and define $$\theta_{\lambda}=\displaystyle\inf_{u\in N_{\lambda}}J_{\lambda}(u),\;\theta^{+}_{\lambda}=\displaystyle\inf_{u\in N^{+}_{\lambda}}J_{\lambda}(u),\;\theta^{-}_{\lambda}=\displaystyle\inf_{u\in N^{-}_{\lambda}}J_{\lambda}(u).$$
%Then, we have the following.
\begin{lemma} If  $0<|\lambda|<\lambda_{0}$, then $$\theta_{\lambda}\leq \theta^{+}_{\lambda}<0\;\;\mbox{and}\;\;\theta^{-}_{\lambda}>d_{0}$$ for some $d_{0}>0$ depending on p, q, $p^{\ast}$, K, $\lambda$, $S_{q}$ and  $S_{p^{\ast}}$.
\end{lemma}
\begin{proof}
Let $u\in N^{+}_{\lambda}$. Then, from (\ref{6}) we have
$$\frac{p-q}{p^{\ast}-q}\|u\|^{p}>\int_{\Omega}F(x,u)dx.$$
So
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
  J_{\lambda}(u) &=& \frac{q-p}{pq}\|u\|^{p}+\frac{p^{\ast}-q}{p^{\ast}q}\int_{\Omega}F(x,u)dx \\
   &<& \left(\frac{q-p}{pq}+\frac{p^{\ast}-q}{p^{\ast}q}\frac{p-q}{p^{\ast}-q}\right)\|u\|^{p}\\
   &=& -\frac{(p-q)(p^{\ast}-p)}{pqp^{\ast}}\|u\|^{p}<0.
\end{eqnarray*}
Thus, from the definition of  $\theta_{\lambda}$ and $\theta^{+}_{\lambda}$, we can deduce that $\theta_{\lambda}\leq \theta^{+}_{\lambda}<0.$\\
Now, let $u\in N^{-}_{\lambda}$. Then, using (\ref{1}) and (\ref{2}) we obtain $$\frac{p-q}{p^{\ast}-q}\|u\|^{p}<\int_{\Omega}F(x,u)dx\leq KS_{p^{\ast}}^{-\frac{p^{\ast}}{p}}\|u\|^{p^{\ast}},$$
this implies that
\begin{equation}\label{13}
    \|u\|>\left(\frac{p-q}{p^{\ast}-q}\frac{S_{p^{\ast}}^{\frac{p^{\ast}}{p}}}{K}\right)^{\frac{1}{p^{\ast}-p}},\;\;\forall u \in N^{-}_{\lambda}.
\end{equation}
In addition, by (\ref{16}) and (\ref{13})
\begin{eqnarray*}
        J_{\lambda}(u) &\geq& \frac{p^{\ast}-p}{pp^{\ast}}\|u\|^{p}-|\lambda|S_{q}^{-\frac{q}{p}}\frac{p^{\ast}-p}{p^{\ast}q}\|u\|^{q} \\
        &\geq& \|u\|^{q}\left[\frac{p^{\ast}-p}{pp^{\ast}}\|u\|^{p-q}-|\lambda|S_{q}^{-\frac{q}{p}}\frac{p^{\ast}-q}{p^{\ast}q}\right] \\
         &>& \left(\frac{p-q}{p^{\ast}-q}\frac{S_{p^{\ast}}^{\frac{p^{\ast}}{p}}}{K}\right)^{\frac{q}{p^{\ast}-p}}
\left(\frac{p^{\ast}-p}{pp^{\ast}}\left(\frac{p-q}{p^{\ast}-q}\frac{S_{p^{\ast}}^{\frac{p^{\ast}}{p}}}{K}\right)
^{\frac{p-q}{p^{\ast}-p}}-
|\lambda|S_{q}^{-\frac{q}{p}}\frac{p^{\ast}-q}{p^{\ast}q}\right).
\end{eqnarray*}
Thus, since $0<|\lambda|<\lambda_{0}$, we conclude that $J_{\lambda}>d_{0}$ for some $d_{0}>0$. This completes the proof of lemma 7.
\end{proof}
For $u\in E$ with $\int_{\Omega}F(x,u)dx>0$, set $$T=\left(\frac{(p-q)\|u\|^{p}}{(p^{\ast}-q)\int_{\Omega}F(x,u)dx}\right)^{\frac{1}{p^{\ast}-p}}>0.$$
%Then, the following lemma hold.
\begin{lemma}
For each $u\in E$ with $\int_{\Omega}F(x,u)dx>0$, we have \\
(i) If $\lambda\int_{\Omega}a(x)|u|^{q}dx\leq 0$, then there exists unique $t^{-}>T$ such that $t^{-}u \in N^{-}_{\lambda}$ and
$$J_{\lambda}(t^{-}u)=\displaystyle\sup _{t\geq 0}J_{\lambda}(tu).$$
(ii) If $\lambda\int_{\Omega}a(x)|u|^{q}dx> 0$, then there are unique $0< t^{+}<T<t^{-}$ such that $(t^{-}u,t^{+}u)\in N_{\lambda}^{-}\times N_{\lambda}^{+}$ and $$J_{\lambda}(t^{-}u)=\displaystyle\sup _{t\geq 0}J_{\lambda}(tu); \;\;\;\;J_{\lambda}(t^{+}u)=\displaystyle\inf _{0\leq t\leq T}J_{\lambda}(tu).$$
\end{lemma}
\begin{proof}
We fix $u\in E$ with $\int_{\Omega}F(x,u)dx>0$ and we let $$m(t)=t^{p-q}\|u\|^{p}-t^{p^{\ast}-q}\int_{\Omega}F(x,u)dx\;\;\;\mbox{for}\;\;t\geq 0.$$
Then, it is easy to check that $m(t)$ achieves its maximum at $T$. Moreover,
 \begin{eqnarray*}
 m(T) &=& \|u\|^{q}\left[(\frac{p-q}{p^{\ast}-q})^{\frac{p-q}{p^{\ast}-q}}-(\frac{p-q}{p^{\ast}-q})^{\frac{p^{\ast}-q}{p^{\ast}-p}}\right]
\left(\frac{\|u\|^{p^{\ast}}}{\int_{\Omega}F(x,u)dx}\right)^{\frac{p-q}{p^{\ast}-p}} \\
 &\geq& \|u\|^{q}(\frac{p^{\ast}-p}{p^{\ast}-q})\left(\frac{(p^{\ast}-q)S_{p^{\ast}}^{\frac{p^{\ast}}{p}}}{K(p-q)}\right)^{\frac{p-q}{p^{\ast}-p}}.
 \end{eqnarray*}
(i) We suppose that $\lambda\int_{\Omega}a(x)|u|^{q}dx\leq 0$. Since $m(0)=0,\;m(t)\rightarrow -\infty$ as $t\rightarrow \infty$, $m'(t)>0$ for $t<T$ and $m'(t)<0$ for $t>T$. There is a unique $t^{-}>T$ such that $m(t^{-})=\lambda \int_{\Omega}a(x)|u|^{q}dx\leq 0.$\\
Now, it follows from (\ref{14}) and (\ref{9}) that
$$\phi'_{\lambda}(t^{-}u)t^{-}u=(t^{-})^{1+q}m'(t^{-})<0$$ and $$J'_{\lambda}(t^{-}u)t^{-}u=(t^{-})^{q}\left(m(t^{-})-\lambda\int_{\Omega}a(x)|u|^{q}dx\right)=0.$$
Hence, $t^{-}u\in N_{\lambda}^{-}$. On the other hand, it is easy to see that   $$\frac{d^{2}}{dt^{2}}J_{\lambda}(tu)<0\;\;\mbox{for}\;\;t>T\;\;\mbox{and}\;\;\frac{d}{dt}J_{\lambda}(tu)=0\;\;\mbox{for}\;\;t=t^{-}.$$
Thus, $J_{\lambda}(t^{-}u)=\displaystyle\sup _{t\geq 0}J_{\lambda}(tu)$.\\
(ii) $\lambda\int_{\Omega}a(x)|u|^{q}dx>0.$ By \textbf{(A)} and (\ref{2}) and the fact that $|\lambda|<\lambda_{0}$ we obtain $$m(0)=0<\lambda \int_{\Omega}a(x)|u|^{q}dx\leq |\lambda|S_{q}^{-\frac{q}{p}}\|u\|^{q}<m(T).$$
Then, there are unique $t^{+}$ and $t^{-}$ such that $0<t^{+}<T<t^{-}$, $m(t^{+})=\lambda \int_{\Omega}a(x)|u|^{q}dx=m(t^{-})$ and $m'(t^{-})<0<m'(t^{+}).$
We have $(t^{-}u,t^{+}u)\in N^{-}_{\lambda}\times N^{+}_{\lambda},$ and
 $$\left\{
 \begin{array}{ll}
 J_{\lambda}(t^{+}u) \leq J_{\lambda}(t u) \leq J_{\lambda}(t^{-}u)\;\;\;\forall\; t\in [t^{+},t^{-}], \\
 J_{\lambda}(t^{+}u) \leq J_{\lambda}(t u)\;\;\forall \;0\leq t \leq t^{+}.
 \end{array}
 \right.$$
Thus, $$J_{\lambda}(t^{-}u)=\displaystyle\sup _{t\geq 0}J_{\lambda}(tu) \;\;\mbox{and}\;\;J_{\lambda}(t^{+}u)=\displaystyle\inf _{0\leq t\leq T}J_{\lambda}(tu).$$
This completes the proof of lemma 8.
\end{proof}
For each $u \in E$ with $\lambda \int_{\Omega}a(x)|u|^{q}dx>0$, let $$\widetilde{T}=\left(\frac{\lambda (p^{\ast}-q) \int_{\Omega}a(x)|u|^{q}dx}{(p^{\ast}-p)\|u\|^{p}}\right)^{\frac{1}{p-q}}>0.$$
Then we have the following lemma.
\begin{lemma}
 For each $u\in E$ with $\lambda\int_{\Omega}a(x)|u|^{q}dx>0$, we have \\
(i) If $\int_{\Omega}F(x,u)dx\leq 0$, then there exists a unique $0<t_{+}<\widetilde{T}$ such that $t^{+}\in N^{+}_{\lambda}$ and
$$J_{\lambda}(t^{+}u)=\displaystyle\inf _{t\geq 0}J_{\lambda}(tu).$$
(ii) If $\int_{\Omega}F(x,u)dx> 0$, then there are unique $0< t^{+}<\widetilde{T}<t^{-}$ such that $(t^{-}u,t^{+}u)\in N_{\lambda}^{-}\times N_{\lambda}^{+}$ and $$J_{\lambda}(t^{-}u)=\displaystyle\sup _{t\geq 0}J_{\lambda}(tu); \;\;\;\;J_{\lambda}(t^{+}u)=\displaystyle\inf _{0\leq t\leq \widetilde{T}}J_{\lambda}(tu).$$
\end{lemma}
\begin{proof}
For  $u\in E$ with $\lambda\int_{\Omega}a(x)|u|^{q}dx>0$, we can take $$\widetilde{m}(t)=t^{p-p^{\ast}}\|u\|^{p}-\lambda t^{q-p^{\ast}}\int_{\Omega}a(x)|u|^{q}dx\;\;\;\mbox{for}\;\;t>0,$$
and similar to the argument in lemma 8, we obtain the results of lemma 9.
\end{proof}
\begin{proposition}There exist minimizing sequences $\{u_{n}^{�}\}$ in $N_{\lambda}^{�}$ such that $$J_{\lambda}(u_{n}^{�})=\theta_{\lambda}^{�}+\circ (1)\;\;\mbox{and}\;\;J'_{\lambda}(u_{n}^{�})=\circ (1)\;\;\mbox{in}\;\;E^{*}.$$
\end{proposition}
\begin{proof}
The proof is almost the same as that in Wu[\cite{R9}, Proposition 9] and is omit here.
\end{proof}
\section{Proof of our result}
Throughout this section,  the $L^{s}$ norm is denoted  by $\|.\|_{s}$ for $1\leq s\leq \infty$, $\rightarrow$ (respectively $\rightharpoonup$) denotes strong (respectively weak) convergence and we assume that the parameter $\lambda$ satisfies $0 < |\lambda| <\lambda_{0}.$
Then we have the following results.
\begin{teo}
 If $0<|\lambda|<\lambda_{0}$, then, problem (1) has a positive solution $u^{+}_{0}$ in $N^{+}_{\lambda}$
 such that $$J_{\lambda}(u^{+}_{0})=\theta_{\lambda}=\theta^{+}_{\lambda}.$$
\end{teo}
\begin{proof}
By Proposition 1, there exists a minimizing sequence
$(u^{+}_{n})_{n}$ for $J_{\lambda}$ on $N^{+}_{\lambda}$ such that
\begin{equation}\label{15}
 J_{\lambda} (u^{+}_{n}) =\theta^{+}_{\lambda} +o(1) \;\;\mbox{and}\;\; J'_{\lambda} (u^{+}_{n}) =o(1)\;\;\mbox{ in}\;\; E^{*}.
\end{equation}
Then by Lemma 3, there exists a subsequence $(u_{n})_{n}$ and $u^{+}_{0}$  in $E$ such that
\begin{equation}\label{24}
\left\{
    \begin{array}{ll}
       u_{n}\rightharpoonup u^{+}_{0}\;\;\mbox{weakly in}\;\;E, \\
      u_{n}\rightarrow u^{+}_{0}\;\;\mbox{strongly in}\;\;L^{q}(\Omega)\;\;\mbox{and in}\;\;L^{p^{\ast}}(\Omega).\;\;\;\;\;\;
    \end{array}
  \right.
\end{equation}
This implies that $\int_{\Omega}a(x)|u_{n}|^{q}dx\rightarrow \int_{\Omega}a(x)|u^{+}_{0}|^{q}dx$ as $n\rightarrow \infty$.\\
Next, we will show that $$\int_{\Omega}F(x,u_{n})dx\rightarrow \int_{\Omega}F(x,u^{+}_{0})dx\;\;\mbox{as}\;\;n\rightarrow \infty.$$
By lemma 1, we have $$\frac{\partial F(x,u_{n})}{\partial u}\in L^{p}(\Omega)\;\;\mbox{and}\;\;\frac{\partial F(x,u_{n})}{\partial u}\rightarrow \frac{\partial F(x,u_{0}^{+})}{\partial u}\;\;\mbox{in}\;\;L^{p}(\Omega).$$
On the other hand, it follows from the H\"{o}lder inequality, that
\begin{eqnarray*}
  \int_{\Omega}\left|u_{n}\frac{\partial F(x,u_{n})}{\partial u}-u_{0}^{+}\frac{\partial F(x,u_{0}^{+})}{\partial u}\right|dx &\leq& \int_{\Omega}|(u_{n}-u_{0}^{+})\frac{\partial F(x,u_{n})}{\partial u}|dx+\int_{\Omega}|u_{0}^{+}|\left|\frac{\partial F(x,u_{n})}{\partial u}-\frac{\partial F(x,u_{0}^{+})}{\partial u}\right|dx \\
   &\leq& \|u_{n}-u_{0}^{+}\|_{p^{\ast}}\|\frac{\partial F(x,u_{n})}{\partial u}\|_{p}+\|u_{0}^{+}\|_{p^{\ast}}\|\frac{\partial F(x,u_{n})}{\partial u}-\frac{\partial F(x,u_{0}^{+})}{\partial u}\|_{p}\\
&\rightarrow& 0\;\;\mbox{as} \;\;n\rightarrow \infty.
\end{eqnarray*}
Hence, $\int_{\Omega}F(x,u_{n})dx \rightarrow \int_{\Omega}F(x,u_{0}^{+})dx$ as $n\rightarrow \infty.$\\
By (\ref{15}) and (3.2) it is easy to prove that $u_{0}^{+}$ is a weak solution of (1).\\
Since $$J_{\lambda}(u_{n})=\frac{p^{\ast}-p}{pp^{\ast}}\|u\|^{p}-\lambda \frac{p^{\ast}-q}{qp^{\ast}} \int_{\Omega}a(x)|u_{n}|^{q}dx\geq -\lambda \frac{p^{\ast}-q}{qp^{\ast}} \int_{\Omega}a(x)|u_{n}|^{q}dx$$
and by (3.1) and lemma 7 , $J_{\lambda}(u_{n})\rightarrow \theta_{\lambda}<0$ as $n\rightarrow \infty$, letting $n\rightarrow \infty$, we see that
\begin{equation}\label{0}
    \lambda \int_{\Omega}a(x)|u_{0}^{+}|^{q}dx>0.
\end{equation}
Now, we aim to prove  that $u_{n} \rightarrow u_{0}^{+}$ strongly in E and $J_{\lambda}(u_{0}^{+})=\theta_{\lambda}.$\\
using the fact that $u_{0}^{+} \in N_{\lambda}$ and by Fatou's lemma, we get
\begin{eqnarray*}
   \theta_{\lambda}&\leq& J_{\lambda}(u_{0}^{+})=\frac{1}{p}\|u_{0}^{+}\|^{p}-\frac{1}{p^{\ast}}\int_{\Omega}F(x,u_{0}^{+})dx-\frac{\lambda}{q}\int_{\Omega}a(x)|u_{0}^{+}|^{q}dx \\
   &\leq& \displaystyle\liminf_{n \rightarrow\infty}\left(\frac{1}{p}\|u_{n}\|^{p}-\frac{1}{p^{\ast}}\int_{\Omega}F(x,u_{n})dx-\frac{\lambda}{q}\int_{\Omega}a(x)|u_{n}|^{q}dx \right) \\
   &\leq& \displaystyle\liminf_{n \rightarrow\infty}J_{\lambda}(u_{n})=\theta_{\lambda}
\end{eqnarray*}
This implies that $$J_{\lambda}(u_{0}^{+})=\theta_{\lambda}\;\;\mbox{and}\;\;\displaystyle\lim_{n \rightarrow\infty}\|u_{n}\|^{p}=\|u_{0}^{+}\|^{p}.$$
Let $\widetilde{u}_{n}=u_{n}-u_{0}^{+}$, then by Br\'ezis-Lieb lemma \cite{R7} we obtain $$\|\widetilde{u}_{n}\|^{p}=\|u_{n}\|^{p}-\|u_{0}^{+}\|^{p}.$$
Therefore, $u_{n} \rightarrow u_{0}^{+}$ strongly in E.\\
Moreover, we have $u_{0}^{+} \in N^{+}_{\lambda}$. In fact, if $u_{0}^{+} \in N^{-}_{\lambda}$ then, there exist $t_{0}^{+}, t_{0}^{-}$ such that $t_{0}^{-} u_{0}^{+} \in N^{-}_{\lambda}$ and $t_{0}^{+} u_{0}^{+} \in N^{+}_{\lambda}$. In particular we have $t_{0}^{+}<t_{0}^{-}=1.$ Since
$$\frac{d^{2}}{dt^{2}}J_{\lambda}(t_{0}^{+}u_{0}^{+})>0\;\;\mbox{and}\;\;\frac{d}{dt}J_{\lambda}(t_{0}^{+}u_{0}^{+})=0\;\;\mbox{for}\;\;t=t^{-}.$$
there exists $t_{0}^{+}< \widetilde{t}<t_{0}^{-}$ such that $J_{\lambda}(t_{0}^{+}u_{0}^{+})<J_{\lambda}(\widetilde{t}u_{0}^{+})$. By Lemma 9, we have $$J_{\lambda}(t_{0}^{+}u_{0}^{+})<J_{\lambda}(\widetilde{t}u_{0}^{+})\leq J_{\lambda}(t_{0}^{-}u_{0}^{+})=J_{\lambda}(u_{0}^{+})$$
which is a contraduction.\\
Finally, by (\ref{0}) we may assume that $u_{0}^{+}$ is a nontrivial solution of problem (1).\\
\end{proof}
\begin{teo} If $0<|\lambda|<\lambda_{0}$, then, problem (1) has a positive solution $u^{-}_{0}$ in $N^{-}_{\lambda}$
 such that $$J_{\lambda}(u^{-}_{0})=\theta^{-}_{\lambda}.$$
\end{teo}
\begin{proof}
By Proposition 1, there exists a minimizing sequence $\{u_{n}\}$ for $J_{\lambda}$ on $N^{-}_{\lambda}$ such that
\begin{equation}\label{22}
J_{\lambda} (u_{n}) =\theta^{-}_{\lambda} +o(1) \;\;\mbox{and}\;\; J'_{\lambda} (u_{n}) =o(1) \;\mbox{in}\; E^{*}
\end{equation}
and
\begin{equation}\label{26}
\left\{
    \begin{array}{ll}
       u_{n}\rightharpoonup u^{-}_{0}\;\;\mbox{weakly in}\;\;E, \\
      u_{n}\rightarrow u^{-}_{0}\;\;\mbox{strongly in}\;\;L^{q}(\Omega)\;\;\mbox{and in}\;\;L^{p^{\ast}}(\Omega).
    \end{array}
  \right.
\end{equation}
Moreover, by (\ref{6}) we obtain
\begin{equation}\label{21}
\int_{\Omega}F(x,u_{n})dx>\frac{p-q}{p^{\ast}-q}\|u_{n}\|^{p}.
\end{equation}
So, by (\ref{13}) and (\ref{21}) there exists a positive constant $\widetilde{C}$ such that
$$\int_{\Omega}F(x,u_{n})>\widetilde{C}.$$
This implies that
\begin{equation}\label{23}
\int_{\Omega}F(x,u^{-}_{0})\geq\widetilde{C}.
\end{equation}
By (\ref{22}) and (\ref{26}), we obtain clearly that $u^{-}_{0}$ is a weak solution of (1).\\
Now, we prove that $u_{n} \rightarrow u^{-}_{0}$ strongly in E. Supposing otherwise, then $$\|u^{-}_{0}\|<\displaystyle\liminf_{n\rightarrow \infty}\|u_{n}\|.$$
By lemma 8, there is a unique $t^{-}_{0}$ such that $t^{-}_{0}u^{-}_{0} \in N^{-}_{\lambda}$. Since $u_{n}\in N^{-}_{\lambda}$, $J_{\lambda}(u_{n})\geq J_{\lambda}(tu_{n})$ for all $t\geq 0$, we have $$J_{\lambda}(t^{-}_{0}u^{-}_{0})<\displaystyle \lim_{n\rightarrow \infty}J_{\lambda}(t^{-}_{0}u_{n})\leq \displaystyle \lim_{n\rightarrow \infty}J_{\lambda}(u_{n})=\theta^{-}_{\lambda}$$
and this is contraduction. Hence $u_{n} \rightarrow u^{-}_{0}$ strongly in E.\\ This imply that $$J_{\lambda}(u_{n})\rightarrow J_{\lambda}(u^{-}_{0})=\theta^{-}_{\lambda}\;\;\mbox{as}\;\;n\rightarrow \infty.$$
By lemma 4 and (\ref{23}) we may assume that $u^{-}_{0}$ is a nontrivial solution of problem (1).\\
\end{proof}
Now, Let us proof Theorem 1: By Theorems 2 and 3, , we conclude that there exists  $u^{+}_{0} \in N^{+}_{\lambda}$ and $u^{-}_{0}\in N^{-}_{\lambda}$ such that
$$\displaystyle J_\lambda (u^+_0)=\inf_{u\in N^+_\lambda }J_{\lambda}(u)\;\;\mbox{and}\;\; \displaystyle J_\lambda (u^-_0)=\inf_{u\in N^-_\lambda }J_{\lambda}(u).$$
Since $N^{-}_{\lambda}\cap N^{+}_{\lambda}=\emptyset$, this implies that $u^{-}_{0}$ and  $u^{+}_{0}$ are distinct.

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