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\fancyhead[CE]{F. M. Yaghoobi and J. Shamshiri} 
\fancyhead[CO]{Existence of solution for a class of fractional problems with ...}



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
Existence of solution for a class of fractional problems with sign-changing functions\\}
%{\bf Existence of solution for a class of fractional problems} 


\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Farajollah Mohammadi Yaghoobi$^*$}\vspace*{-2mm}\\
\vspace{2mm} {\small  Hamedan Branch, Islamic Azad University} \vspace{2mm}

{\bf  Jamileh Shamshiri\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small   Tabaran Institute of Higher Education} \vspace{2mm}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} Here we study the existence and multiplicity of solutions for the following fractional problem
 $$ (-\Delta)_p^s u+a(x) |u|^{p-2} u= f(x,u), $$
 with the Dirichlet boundary condition $u=0$ on $\partial\Omega$ 
 where $\Omega$ is a bounded domain with smooth boundary, $p\geq 2$,  $s\in(0,1)$ and $a(x)$ is a sign-changing function.
 Moreover, we consider two different assumptions on the function $f(x,u)$,
 including the cases of nonnegative  and sign-changing function.
 \end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 35B38;  35B38; 35D30.

\noindent{\bf Keywords and Phrases:} Critical points, Fractional partial differential equations,  Weak solutions,  Nehari manifold,  Fibering map.
\end{quotation}

\section{Introduction and preliminary results}
\label{intro} % It is advised to give each section and subsection a unique label.
The topics related to the existence and multiplicity of solutions for fractional elliptic problems are very extensive. Also, fractional problems naturally arise in many different branch of science such as optimization \cite{Duvaut-Lions}, conservation laws \cite{Biler-Karch-Woyczynski},  water waves \cite{Craig-Groves,Llave-Valdinoci}, quantum mechanics \cite{Fefferman-Llave}, finance \cite{Cont-Tankov}, minimal surfaces \cite{Caffarelli-Roquejoffre-Savin,Caffarelli-Valdinoci}, phase transitions \cite{Garroni-Palatucci,Sire-Valdinoci}
 and other sciences (see also \cite{Ambrosio-Isernia-2018,Caffarelli-Mellet-Sire,Chermisi-Valdinoci-2009,Chermisi-Valdinoci-2010,Gonzalez-Monneau,Kurzke,Liejun Shen-2018,Mellet-Mischler-Mouhot,Savin-Valdinoci,Silvestre-2005,Silvestre-2007,Vázquez}).
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In this paper, we investigate the existence and multiplicity of solutions for the following  fractional problem 
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%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
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\begin{equation}\label{eP}
 \left\{\begin{array}{cc}\begin{split}
&(-\Delta)_p^s u+a(x) |u|^{p-2} u= f(x,u),  &x\in\Omega,\\
& u=0, & x\in \partial\Omega,
\end{split}\end{array}\right.
 \end{equation}
%
%
%
where $\Omega$ is a bounded subset of $\mathbb{R}^n$, $n>ps$ with $s\in(0,1)$, $p\geq 2$, $a(x)\in L^\infty(\Omega)$ is a sign-changing function and 
%
%
%      equation (1.1a)
%
\begin{equation}\label{eq:1.1a}
(-\Delta)_p^{s}u(x):=2\lim_{\epsilon\rightarrow 0}\int_{\mathbb{R}^n\setminus B_\epsilon(x)}\frac{|u(y)-u(x)|^{p-2}\big(u(y)-u(x)\big)}{|x-y|^{n+ps}}dy.
\end{equation}
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%
 %
In addition, one of the following assumptions is satisfied:
\begin{itemize}
\item[(f1)] $f(x,u)=b(x)|u|^{q-2}u$, where $b(x)\in L^\infty(\Omega)$,  $b(x)\geq 0$ a.e. in $\Omega$ and $p<q<p^*_s$ where $p^*_s=\frac{np}{n-ps}$ is the fractional Sobolev exponent,
\item[(f2)] $f(x,u)=b(x)|u|^{q-2}u+\lambda g(x,u)-h(x)|u|^{r-2}u$, where $\lambda>0$, $2\leq r\leq p<q<p^*_s$, $ b(x) \in L^\infty(\Omega)$ which may change sign and $h(x)\in C(\bar\Omega)$ is a nonnegative function.
\end{itemize}
 %
%

Recently a great deal of attention has been focused on the study of existence and multiplicity of solutions for fractional differential equations. In particular, in the case of $f(x,u)=\lambda b(x)u^q$,  the problem (\ref{eP}) has  been studied by some authors and the existence of multiple positive solutions has been established. For instance, Brown and Wu in \cite{Brown- Wu-2007} considered the following problem 
$$-\Delta_p u=\lambda a(x) |u|^q+b(x)u^p, \quad \quad            x\in \Omega,$$
with Dirichlet boundary condition, where $\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$, $\lambda>0$, $q<1<p<\frac{n+2}{n-2}$ and $a,b:\Omega\rightarrow\mathbb{R}$ are smooth
functions which may change sign on $\Omega$. They proved the existence of at least two positive solutions by using the Nehari manifold and fibering maps. 
 Also, Barrios et al. in \cite{Barrios-Colorado-Servadei-Soria-2015} obtained the existence and multiplicity of solutions for the following fractional differential equation
%
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\begin{equation*}
 (-\Delta)^s u=\lambda u^q+ u^{2^*_s-1},  \quad x\in\Omega,\\
 \end{equation*}
%
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%
with Dirichlet boundary condition, where $\Omega\subset\mathbb{R}^n $  is a regular bounded domain, $\lambda>0$, $0<s<1$, $n>2s$ and $(-\Delta)^s $ denotes the fractional Laplace operator defined by
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\begin{equation*}
-(-\Delta)^{s}u(x)=2\int_{\mathbb{R}^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}dy, \qquad x\in \mathbb{R}^n.
\end{equation*}
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Moreover, G. Ning et al. in \cite{ning-wang-zhang-2018} proved the existence, multiplicity and bifurcation results for the following problem with Dirichlet boundary condition
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\begin{equation*}
 (-\Delta)^s u=\lambda |u|^{q-2}u+ \frac{|u|^{p^*_{s,\alpha}-2}u}{|x|^\alpha},  \quad x\in\Omega,\
  \end{equation*}
%
%
 where $p\in(1,\infty)$, $0<s<1$, $\Omega$ is a bounded domain containing the origin in $\mathbb{R}^n$ with Lipschitz boundary,   $0\leq\alpha<ps<n$ and $p^*_{s,\alpha}=\frac{(n-\alpha)p}{n-ps}$ is the fractional Hardy-Sobolev exponent.
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As well as in \cite{Saiedinezhad-2018} Saiedinezhad, by using the Nehari manifold with variational arguments, the existence of solutions for the following semilinear elliptic equation was studied
 $$\Delta^2u-a(x)\Delta u=b(x)|u|^{p-2}u,$$ 
with Navier boundary condition $\Delta u=u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$  with smooth boundary and $2<p<2^*=\frac{2n}{n-2}$. In this paper, the author considered two different
assumptions on the potentials $a(x)$ and $b(x)$ including the case of sign-changing weights.
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Motivated by the above achievements, we will use variational methods to the study of existence and multiplicity of solutions for problem  (\ref{eP}),
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and for this purpose, we consider the fractional Sobolev space $W_0^{s,p}(\Omega)$
%
%
%
%
%\begin{equation*}
%W^{s,p}(\Omega):=\bigg\{u| u:\mathbb{R}^n\rightarrow \mathbb{R}, \ u|_\Omega\in L^p(\Omega), \  \int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|%^{n+ps}}dxdy<\infty\bigg\},
%\end{equation*}
%
%
with the norm 
%
\begin{equation}\label{eq:1.2a}
\|u\|:=\|u\|_{W_0^{s,p}(\Omega)}=\bigg(\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy\bigg)^{\frac{1}{p}}.
\end{equation}
%
%
For the convenience of the reader we repeat the relevant material from \cite{Chen-Deng-2016} without proofs.
Assume 
$$X=\bigg\{u| u:\mathbb{R}^n\rightarrow \mathbb{R}, \ u|_\Omega\in L^p(\Omega), \  \int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy<\infty\bigg\},$$
 with the norm 
 $\|u\|_X=\bigg(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy\bigg)^{\frac{1}{p}}$, 
 where 
 $Q:=\mathbb{R}^{2n}\backslash (\mathcal{C}\Omega\times\mathcal{C}\Omega)$  and $\mathcal{C}\Omega=\mathbb{R}^n\backslash \Omega$.  
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Also set $X_0$ denotes the closure of $C_0^\infty(\Omega)$ in $X$. By Theorems in \cite{Di Nezza-Palatucci-Valdinoci} the space $X_0$ is a Hilbert space with the scalar product defined for any $u,v\in X_0$ as
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\begin{equation*}\label{eq:1.2b}
\langle u,v \rangle =\int_Q \frac{|u(x)-u(y)|^{p-1}(v(x)-v(y))}{|x-y|^{n+ps}}dxdy,
\end{equation*}
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and  the norm
\begin{equation*}\label{eq:1.2b}
\| u\|_{X_0} :=\bigg(\int_Q \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+ps}}dxdy\bigg)^\frac{1}{p},
\end{equation*}
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which is equivalent to the equation defined in (\ref {eq:1.2a}). Moreover, based on the results found in \cite{Goyal-Sreenadh-2013,Di Nezza-Palatucci-Valdinoci}, it can be said that the embedding $X_0\hookrightarrow L^q(\Omega)$ is continous for any $q\in [1,p^*]$ and compact whenever $q\in [1,p^*)$.
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Thus,  there exists a positive constant $S_p$ such that
$\| u\|_{L^p(\Omega}\leq S_p\times\|  u\|_{X_0}$. Indeed, the sharp constant $S_p$  is equal  to
$\frac{1}{\mu_p}$, where
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 \begin{equation}\label{eq:1.3b}
 \mu_p:=\inf\{\frac{\|u\|_{X_0}}{\| u\|_{L^p(\Omega)}}: \, 0\neq u\in X_0\}.
 \end{equation}
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\section{Main results}
\label{sec:2}
The main  results of this paper are in
 two parts. In the third section, we consider and solve problem (\ref{eP})
by assuming condition {\rm (f1)}. Therefore,  according to the basic variational arguments, we know that the weak solutions of
\eqref{eP} under assumption {\rm (f1)}, is corresponded to the local minimizer of
\begin{equation}\label{eq:1.3c}
I(u)=\frac{1}{p}M(u)+\frac{1}{p}A(u)-\frac{1}{q}B(u),
\end{equation}
where $I:X_0\rightarrow \mathbb{R}$ is the associated  Euler Lagrange functional, $M(u):=\|u\|_{X_0}^p$,  $A(u):=\int_\Omega a(x)|u|^pdx$ and $B(u):=\int_\Omega b(x)|u|^qdx$. 

Also in section 4, where condition {\rm (f2)} is satisfied, we have the following problem
%
\begin{equation}\label{eP2}
 \left\{\begin{array}{cc}\begin{split}
&(-\Delta)_p^s u+a(x) |u|^{p-2}u= b(x) |u|^{q-2}u+\lambda g(x,u)-h(x)|u|^{r-2}u,  &x\in\Omega,\\
& u=0, & x\in \partial\Omega.
\end{split}\end{array}\right.
 \end{equation}
%
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 The Euler Lagrange functional associated with this problem is $\tilde{I}:X_0\rightarrow \mathbb{R}$ such that
  %
 %
\begin{equation}\label{eq2.4}
\tilde{I}(u)=\frac{1}{p}\big(M(u)+A(u)\big)-\frac{1}{q}B(u)-\lambda\int_\Omega G(x,|u|)dx+\frac{1}{r}H(u),
\end{equation}
%
%
where $G(x,u):=\int_0^u g(x,s)ds$ and $H(u):=\int_{\Omega}h(x)|u|^rdx$. 
%
%
 Also we know that if $\tilde{I}(u)$ denotes the energy functional corresponding to a problem, then all of the critical points of $\tilde{I}(u)$ must lie in the set
$N:=\{u; \langle {\tilde{I}'}(u),u\rangle=0\}$, which is known as the Nehari manifold
(see \cite{Nehari,Willem}). On the other hand, the fibering map 
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\begin{equation}\label{eq2.5c}
\varphi_u(t):= \tilde{I}(tu)=\frac{t^p}{p}\big(M(u)+A(u)\big)-\frac{t^q}{q}B(u)-\lambda \int_\Omega G(x,t|u|)dx+\frac{t^r}{r}H(u),
\end{equation}
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 is closely linked to the Nehari manifold, i.e., $\varphi'_u(1)=0$ if and only if $u\in N$, (see \cite{Brown-zhang-2003,Derabek-pohozaev-1997}).  
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 %
  %
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 So it is reasonable to divide the Nehari manifold into three parts corresponding to local minima, local maxima and inflction points of the 
critical points of $\varphi'_u(t)$, and hence we define
$N^+:=\{u\in N, \varphi''_u(1)>0\}$,
$N^-:=\{u\in N, \varphi''_u(1)<0\}$ and
$N^0:=\{u\in N, \varphi''_u(1)=0\}$.
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 %


Besides,  we set 
 %
 %
\begin{equation}\label{eq2.5}
a^+:=\operatorname{ess\,sup}\{a(x), x\in \Omega\},
\end{equation}
and
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%
\begin{equation}\label{eq2.6}
\tilde{A}(u):=M(u)+A(u)=\|u\|_{X_0}^p+\int_{\Omega}a(x)|u|^pdx.
\end{equation}
%
%
In the sequel we need the following lemma:
%

\begin{lemma} \label{lem4}
If $a^+<\mu_p$, then there exists  $\delta_1>0$ such that for every $u\in X_0$,
$\tilde{A}(u)\geq \delta_1\| u\|_{X_0}^p$.
\end{lemma}

{\bf Proof.}
If $\int_\Omega a(x)|u|^pdx \geq0$, then the  lemma is obvious;
otherwise, it would be proved by contradiction, supposing that $\int_\Omega a(x)|u|^pdx<0$. 
If for every  $\delta>0$, there exists $u\in X_0$ such that $\tilde{A}(u)<\delta\| u\|_{X_0}^p$,
 it can be deduced that
\begin{align}\label{3}
(1-\delta)\| u\|_{X_0}^p<\int_\Omega -a(x) |u|^pdx<a^+\int_\Omega |u|^pdx.
\end{align}
Now, by taking $\delta<1-\frac{a^+}{\mu_p}$ we get 
$\frac{a^+}{1-\delta}<\mu_p$
which  leads to a  contradiction with (\ref{eq:1.3b}).$\Box$

%
%
\begin{remark}\label{rem.1}
According to lemma \ref{lem4} we know that if $a^+<\mu_p$ then $\tilde{A}(u)\geq \delta_1\| u\|_{X_0}^p$ for $\delta_1>0$; on the other hand $\tilde{A}(u)\leq \|u\|_{X_0}^p+\|a\|_\infty S_p^p \|u\|_{X_0}^p$. Therefore, in the sequel for $a^+<\mu_p$, we consider $X_0$ with the following norm: 
%{
\begin{equation}\label{3,1}
\|u\|_{\tilde{A}}:=\big(\tilde{A}(u)\big)^\frac{1}{p}=\bigg(\|u\|_{X_0}^p+\int_{\Omega}a(x)|u|^pdx\bigg)^\frac{1}{p}.
\end{equation}
\end{remark}
%   
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 The main results in this paper are the following theorems.
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 %
 %
 %
 \begin{theorem} \label{thm2.2} 
  Suppose that $f(x,u)$  satisfies  condition {\rm (f1)} and $a^+<\mu_p$, then problem \eqref{eP}
admits at least one weak solution in $X_0$.
 \end{theorem}
 %
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 %
 %
 %
%               
%
\begin{theorem}\label{thm2.6} 
Assume $a^+<\mu_p$, then:
\\
(i). If $f(x,u)$  satisfies  condition {\rm (f2)},  then there exists $\lambda^*$ such that for  $0<\lambda<\lambda^*$,  $\tilde{I}$ admits a minimizer on $N^{+}$ which is a nontrivial weak solution of problem (\ref{eP2}).
\\
(ii). If $f(x,u)$  satisfies  condition {\rm (f2)}, then there exists $\lambda^{**}$ such that for  $0<\lambda<\lambda^{**}$,  there exists a minimizer of $\tilde{I}$ on $N^-$ which is a nontrivial weak solution of problem (\ref{eP2}).
\end{theorem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Sect.~\ref{subsec:1}.
\section{Proof of Theorem \ref{thm2.2}}

In this section, we consider problem \eqref{eP} such that $f(x,u)$  satisfies condition {\rm (f1)}, using (\ref{eq:1.3c}) for every $u\neq 0$, $I(tu)\hookrightarrow-\infty$  as $t\hookrightarrow\infty$.
Thus, $I(u)$ is not bounded below and so the minimizing process on the hole space $X_0$ may not be possible. If for every $\alpha\in\mathbb{R}$, we let
 $$
B_\alpha:=\{u\in X_0: \int_\Omega b(x)| u|^qdx=\alpha\},
$$
%
%
then, for every $u\in B_\alpha$, by using  remark \ref{rem.1} we have  $I(u)=\frac{1}{p}\|u\|^p_{\tilde{A}}-\frac{1}{q}\alpha$.
Thus, $I|_{B_\alpha}$  is certainly bounded below and the minimizing approach of $I(u)$
on $B_\alpha$  is equivalent to the minimizing approach of $\|u\|^p_{\tilde{A}}$ on $B_\alpha $. Set $\inf_{u\in B_\alpha}\|u\|^p_{\tilde{A}}=:m_\alpha$,
we will show that $m_\alpha$ is obtained by a function, and a multiple of this
function is a minimizer of $I(u)$ and as a result, weak solution of \eqref{eP}.

\begin{lemma} \label{lem2.1}
 For every $\alpha>0$, there exists a nonnegative function $u_\alpha\in B_\alpha$
such that $\|u\|^p_{\tilde{A}}=m_\alpha$.
 \end{lemma}

 {\bf Proof.}
By the  coercivity of $I(u)$ on $B_\alpha$, there exists a
bounded minimizer sequence $\{u_n^{(\alpha)}\}$ for $\chi(u):=\|u\|^p_{\tilde{A}}$ on $B_\alpha$. Therefore, $\{| u_n^{(\alpha)}|\}$ is a minimizer sequence  in $B_\alpha$, so we can suppose  that $u_n^{(\alpha)}(x)\geq0$ a.e. on $\Omega$.
By the reflexivity of $X_0$, there exists a subsequence of $\{u_n^{(\alpha)}\}$, for simplicity is denoted by $\{u_n^{(\alpha)}\}$, which is weakly convergent to
$u_\alpha\in X_0$ ($u_n^{(\alpha)}\rightharpoonup u_\alpha$). So by compact embedding $X_0\hookrightarrow L^q(\Omega)$,  
$\{u_n^{(\alpha)}\}$ is  strongly convergent in $L^q(\Omega)$, and hence
$$
lim_{n\to\infty}\int_\Omega b(x)| u_n^{(\alpha)}|^q dx
=\int_\Omega b(x)| u_\alpha|^qdx,
$$
which means $u_\alpha\in B_\alpha$. If $u_n^{(\alpha)}\not\to u_\alpha$
in $X_0$, then $\|u_\alpha\|^p_{\tilde{A}}<\liminf \|u_n\|^p_{\tilde{A}}=m_\alpha$,
which is a contradiction with $u_\alpha\in B_\alpha$. So $u_n\to u_\alpha$ in
$X_0$ and  $m_\alpha=\inf_{u\in S_\alpha}\|u\|^p_{\tilde{A}}=\| u_\alpha\|^p_{\tilde{A}}$.
$\Box$

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Proof of Theorem \ref{thm2.2}.}
Let   $\chi(u)=\| u\|^p_{\tilde{A}}$, now if $u_\alpha$ is a minimizer
of $\chi(u)$ under  the condition $B(u)=\alpha$, then by Lagrange multiplier theorem,  there exists
$\lambda\in\mathbb{R}$ such that $\chi'(u_\alpha)=\lambda B'(u_\alpha)$, and hence for every $v\in X_0$ we have
%
%
\begin{equation}\label{1}
\langle \chi'(u_\alpha),v\rangle=q\lambda \int_\Omega b(x)| u_\alpha|^{q-2}  u_\alpha v dx.
\end{equation}
By taking $u_\alpha=C w_\alpha$ we get
\begin{equation*}\begin{split}
&C^{p-1}\int_{Q}\frac{|w_\alpha(x)-w_\alpha(y)|^{p-2}(w_\alpha(x)-w_\alpha(y))(v(x)-v(y))}{|x-y|^{n+ps}}dxdy
\\&+C^{p-1}\int_\Omega a(x)| w_\alpha|^{p-2}  w_\alpha v dx
=\frac{q\lambda}{p} C^{q-1}
\int_\Omega b(x)| w_\alpha|^{q-2}  w_\alpha v dx.
\end{split}\end{equation*}
%
%
Now, by assuming $C=(\frac{p}{q\lambda})^{\frac{1}{q-p}}$ we have
%
%
 \begin{equation}\begin{split}\label{1}
&\int_{Q}\frac{|w_\alpha(x)-w_\alpha(y)|^{p-2}(w_\alpha(x)-w_\alpha(y))(v(x)-v(y))}{|x-y|^{n+ps}}dxdy
\\&+\int_\Omega a(x)| w_\alpha|^{p-2}  w_\alpha v dx=
\int_\Omega b(x)| w_\alpha|^{q-2}  w_\alpha v dx,
\end{split}\end{equation}
%
%
consequently, $w_\alpha$ is a weak solution of \eqref{eP} under assumption {\rm (f1)}.
$\Box$

\begin{lemma} \label{lem2.3} \rm
For $\alpha\neq \beta$ the minimizers of $\chi(u)$ on $B_\alpha$ and $B_\beta$
give the same weak solution of \eqref{eP}.
\end{lemma}

{\bf Proof.}
For $\alpha\neq \beta$, we have
\[
B_\alpha=\big\{u\in X_0: \int_\Omega b(x) | u| ^qdx=\alpha\big\}
=\big\{\big(\frac{\alpha}{\beta}\big)^{1/q}v: v\in X_0, \int_\Omega b(x)
| v| ^qdx=\beta  \big\}.
\]
Therefore,
\begin{align}\label{2}
m_\alpha=\inf_{u\in B_\alpha} \|u\|^p_{\tilde{A}}
=\big(\frac{\alpha}{\beta}\big)^{p/q}m_\beta.
\end{align}
So $u_\alpha$ minimizes $\|u\|^p_{\tilde{A}}$ on $B_\alpha$ if and only if
$(\frac{\beta}{\alpha})^{1/q} u_\alpha$ minimizes  $\|u\|^p_{\tilde{A}}$ on $B_\beta$. Indeed,
  by  substituting $C_\alpha=(\frac{\alpha}{m_\alpha})^{\frac{1}{q-p}}$ we have
%
\begin{align*}
w_\alpha
=\frac{1}{C_\alpha} u_\alpha=(\frac{m_\alpha}{\alpha})^{\frac{1}{q-p}}
\big(\frac{\alpha}{\beta}\big)^{1/q}u_\beta
= (\frac{m_\beta}{\beta})^{\frac{1}{q-p}}u_\beta=\frac{u_\beta}{C_\beta}=w_\beta. \Box
\end{align*}

%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of Theorem \ref{thm2.6}}
\label{sec:3}
 %
 In this section, where condition {\rm (f2)} is satisfied, 
we study the existence and multiplicity results for problem (\ref{eP2}). 
 One of the main difficulties in this problem will be the nonlinearity of  $g(x, u)$. To overcome this difficulty we need to restrict $g(x,u)$ to the following conditions:
%
%
%
\begin{itemize}
\item[(g1)]  $g(x,u) \in C^1({\Omega} \times \mathbb{R})$ such that
$g(x,0)\geq0$, $g(x,0)\not\equiv0$ and there exists $\bar{g_1}(x)\in L^\infty(\Omega)$ such
that, $|g_u(x,u)| \leq \bar{g_1}(x)u^{p-2}$
where $(x,u)\in \Omega\times\mathbb{R^+}$.

%
%
 %
\item[(g2)]  For $u\in L^p(\Omega),\, \int_{\Omega}g_u(x,t|u|)u^{2}dx$ has the same sign for every $t\in (0,\infty).$
\end{itemize}
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 %
 %
 %
%%%%%%%%%%%%%%%%%
A typical example of $g(x,u)$ is given by $g(x,u)=\sqrt[4]{(1+u^2)^p}$, for other examples, please refer to \cite{Aghajani-Shamshiri-Yaghoobi}.
%
% 
 
 %
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%  
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%
%                 remark(1.1)
%
%
\begin{remark}\label{rem1.1}
If $g(x,u)$ satisfies  (g1), then there exists $\bar{g_2}(x)\in L^p(\Omega)$ such that $g(x,u)\leq \bar{g_2}(x)(1+u^{p-1})$ and $G(x,u)\leq 2\bar{g_2}(x)(1+u^p)$, for all $(x,u)\in \Omega \times \mathbb{R}^+$.
Moreover, based on the
compactness of the embedding
$X_{0}\hookrightarrow\mathrm{{L}}^{r}(\Omega)$ for
$1\leq r<p^*$ and the fact that the operator $u\longmapsto g(x,u)$ is
continuous, we conclude that the functional
$J(u)=\int_{\Omega}{G(x,u)dx}$ is weakly continuous, i.e., if
$u_{n}\rightharpoonup u$, then $J(u_{n})\rightarrow J(u)$  and moreover the
operator $J'(u)=\int_{\Omega}{g(x,u)udx}$ is weak to strong
continuous, i.e., if
 $u_{n}\rightharpoonup u$, then $J'(u_{n})\rightarrow
 J'(u).$
 \end{remark}

 %
%
 
%
The following lemma shows that minimizers for $\tilde{I}$ on $N$ are usually critical
points for  $\tilde{I}$, as proved by Brown and Zhang in \cite{Brown-zhang-2003}

%        Lemma(2.1)
%
\begin{lemma} \label{lem2.02}
Let $u_{0}$ be a local minimizer for $\tilde{I}(u)$
 on $N$ such that
$u_{0}\notin N^{0},$ then $u_{0}$ is a
critical point of $\tilde{I}(u)$.
\end{lemma}
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%    
%
 %
 
 Motivated by the above lemma, we will get conditions for $N^{0}=\emptyset$
%
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%                     Lemma(2.2)
%
\begin{lemma}\label{lem2.03}
If $a^+<\mu_p$ then there exists $\lambda_{0}>0$ such that for
$0<\lambda<\lambda_{0}$, we have $N^{0}=\emptyset$.
\end{lemma}
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{\bf Proof.} Suppose the other way round, then  for $u\in N^{0}$,  
\label{sec:4}
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using (g1), (\ref{eq2.5c}) and the relation of $\varphi''_u(1)=0$,  we have $(p-1)\|u\|^p_{\tilde{A}}\leq (q-1)\tilde{S}_q^q\|b\|_\infty\|u\|^q_{\tilde{A}}+\|\bar{g_1}\|_\infty \tilde{S}_p^p\lambda\|u\|^p_{\tilde{A}}$, \big(where $\tilde{S}_r$  denotes  the best Sobolev 
constant for the embedding
 of $X_0$ with the norm $\|u\|_{\tilde{A}}$ into $\mathrm{L}^{r}(\Omega)$\big) and hence
 %
 %
 %
%
%
%
%
%               equation(2.10)
%
\begin{equation}\label{2.10}
\|u\|_{\tilde{A}}\geq \Big(\frac{p-1-\|\bar{g_1}\|_\infty \tilde{S}_p^p\lambda}{(q-1)\tilde{S}_q^q\|b\|_\infty}\Big)^{\frac{1}{q-p}}.
\end{equation}
%
%
%
On the other hand, from (\ref{eq2.5c}), (g1), remark \ref{rem1.1} and using the fact that $(q-1)\varphi'_u(1)-\varphi''_u(1)=0$, we obtain
%
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%
\begin{equation*}\begin{split}
(q-p)\|u\|^p_{\tilde{A}}&\leq\lambda
\big(\int_{\Omega}(q-1)g(x,|u|)|u|-g_{u}(x,|u|)u^2\big)dx+(r-q)H(u)\\&\leq 2\lambda(q-1)\|\bar{g_2}\|_\infty|\Omega|+\lambda(2(q-1)\|\bar{g_2}\|_\infty+\|\bar{g_1}\|_\infty)\tilde{S}_p^p\|u\|^p_{\tilde{A}},
\end{split}\end{equation*}
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which concludes
%
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%               equation(2.11)
%
\begin{equation*}
\|u\|_{\tilde{A}}\leq \big(\frac{2(q-1)\lambda\|\bar{g_2}\|_\infty |\Omega|}{q-p-\lambda(2(q-1)\|\bar{g_2}\|_\infty+\|\bar{g_1}\|_\infty)\tilde{S}_p^p}\big)^\frac{1}{p}.
\end{equation*}
%
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Therefore, using (\ref{2.10})  we must have
%
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%
 \begin{equation*}
\big(\frac{p-1-\|\bar{g_1}\|_\infty S\tilde{S}_p^p\lambda}{(q-1)\tilde{S}_q^q\|b\|_\infty}\big)^{\frac{1}{q-p}}\leq
\big(\frac{2(q-1)\lambda\|\bar{g_2}\|_\infty |\Omega|}{q-p-\lambda(2(q-1)\|\bar{g_2}\|_\infty+\|\bar{g_1}\|_\infty)\tilde{S}_p^p}\big)^\frac{1}{p},
\end{equation*}
%
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%
 which is a contradiction for $\lambda$ sufficiently small.
So there exists $\lambda_{0}>0$ such that for
$0<\lambda<\lambda_{0},$ \  $N^{0}=\emptyset.$
$\Box$
%
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%
%
%          Lemma(2.1)

\begin{lemma}\label{lem2.1}\label{lem2.1}
If $a^+<\mu_p$ then there exists $\lambda_1>0$ such that for $\lambda<\lambda_1$,
$\tilde{I}(u)$ is coercive and bounded below on
$N$.
\end{lemma}
%
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%
{\bf Proof.} For $u\in N$, using  (\ref{eq2.4}) and remark \ref{rem1.1} we have 
 %
%
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%
\begin{equation*}\begin{split}
\tilde{I}(u)&=(\frac{1}{p}-\frac{1}{q})\|u\|^p_{\tilde{A}}-\lambda\int_{\Omega}(G(x,|u|)-\frac{1}{q}g(x,|u|)|u|)dx+(\frac{1}{r}-\frac{1}{q})H(u)\\
&\geq (\frac{1}{p}-\frac{1}{q})\|u\|^p_{\tilde{A}}-2\lambda \|\bar{g_2}\|_\infty(1+\frac{1}{q})|\Omega|-2\lambda \|\bar{g_2}\|_\infty \tilde{S}_p^p(1+\frac{1}{q})\|u\|^p_{\tilde{A}}.
\end{split}\end{equation*}
%
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As a result, $\tilde{I}$ is coercive and bounded
below on
$N$ for $0<\lambda<\lambda_1=\frac{q-p}{2p(q+1)\|\bar{g}_2\|_\infty \tilde{S}_p^p}$. 
$\Box$

%
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%
%          lemma(2.5)
%
\begin{lemma}\label{lem2.5}
If $a^+<\mu_p$ then there exists $\lambda_{2}>0$ such that, for $0<\lambda <\lambda_{2}$, $\varphi_u(t)$ takes on positive values for all non-zero $u\in X_{0}$.
\end{lemma}
%
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{\bf Proof.} If $B(u)\leq0$, then using (\ref{eq2.5c}) and by elementary calculus we can show that 
$\varphi_u(t)>0$ for sufficiently large $t$. Suppose there exists $u\in X_{0}$ such that $B(u)>0,$
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%
through using (\ref{eq2.5c}), (\ref{3,1}) and by elementary calculus, $\psi_u(t):=\frac{t^p}{p}\|u\|^p_{\tilde{A}}-\frac{t^q}{q}B(u)$ takes on a maximum at ${t}_{max}=\big(\frac{\|u\|^p}{B(u)}\big)^{\frac{1}{q-p}}$ and so
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%
%
%                 equation(2.18)
%
\begin{equation}\begin{split}\label{eq2.18}
\psi_u(t_{max})
=(\frac{1}{p}-\frac{1}{q})\bigg\{\frac{(\|u\|_{\tilde{A}}^p)^{q}}{(B(u) )^{p}}\bigg\} ^{\frac{1}{q-p}}\geq(\frac{1}{p}-\frac{1}{q})\bigg\{\frac{1}{|b^+\|^p_\infty \tilde{S}_q^{pq} }\bigg\} ^{\frac{1}{q-p}}:=\delta_2>0,
\end{split}\end{equation}
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where ${\delta_2}$ is independent of u. Moreover, for
$1\leq \alpha<p^*$ we have 
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\begin{equation*}\begin{split}
 (t_{max})^{\alpha}\int_{\Omega}|u|^{\alpha}dx
&\leq \tilde{S}_{\alpha}^{\alpha}\bigg(\frac{\|u\|_{\tilde{A}}^p}{B(u)}\bigg)^{\frac{\alpha}{q-p}}(\|u\|_{\tilde{A}}^p)^{\frac{\alpha}{p}}
=\tilde{S}_{\alpha}^{\alpha}\bigg\{\frac{\|u\|_{\tilde{A}}^{pq}}{(B(u))^{p}}
\bigg\}^{\frac{\alpha}{p(q-p)}}\\&=\tilde{S}_{\alpha}^{\alpha}\big(\frac{qp)}{q-p}\big)^{\frac{\alpha}{p}}(\psi_u(t_{max})\big)^{\frac{\alpha}{p}}
=c(\psi_u(t_{max})\big)^{\frac{\alpha}{p}},
\end{split}\end{equation*}
%
%
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hence using  remark \ref{rem1.1}, we conclude that
%
%
%                equation(2.20)
%
\begin{equation*}\begin{split}
\lambda\int_{\Omega}G(x,t_{max} |u|)dx-\frac{1}{r}H(t_{max} |u|)&\leq
2\lambda\|\bar{g_2}\|_\infty\int_{\Omega}\big(1+|{t}_{max}u|^p\big)dx\\&\leq
2\lambda\|\bar{g_2}\|_\infty|\Omega|+c_1\lambda\|\bar{g_2}\|_\infty{\psi_u}({t}_{max}),
\end{split}\end{equation*}
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where $c_1$ is independent of $u$. So from (\ref{eq2.18}) we get
%
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%
\begin{equation*}\begin{split}
\varphi_u(t_{max})&=\psi_u(t_{max})-\lambda \int_\Omega G(x,t_{\max} |u|)dx+\frac{1}{r}H(t_{\max} |u|)\\
&\geq \psi_u(t_{max})\bigg(1-\lambda 2\|\bar{g_2}\|_\infty |\Omega|\big({\psi_u}(t_{max})\big)^{-1}-\lambda c_1\|\bar{g_2}\|_\infty\bigg)\\
&\geq\delta_2\bigg(1-\delta_2^{-1}\lambda 2\|\bar{g_2}\|_\infty|\Omega|-\lambda c_1\|\bar{g_2}\|_\infty\bigg).
\end{split}\end{equation*}
%
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%
Clearly $\varphi_u({t}_{max})>\epsilon>0$, for all nonzero $u$,
provided that $$\lambda<\lambda_{2}:=\frac{1}{2\|\bar{g_2}\|_\infty|\Omega|\delta_2^{-1}+c_1\|\bar{g_2}\|_\infty}.\Box$$


%
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%

Since $\varphi_u(0)\leq0$, so it is clear that if $a^+<\mu_p$ and $0<\lambda <
\lambda_{2}$ then there exists $0<\tau<t_{max}$ such that $\varphi'_u(\tau)>0$, and so we have the following corollary:
%
%
%
%          Corollary(2.2)
%
\begin{corollary}\label{cor2.2}
 (i). If $a^+<\mu_p$, $0<\lambda<\lambda_1$ and $B(u)\leq0$ for $u\in X_0\setminus
\{0\},$ then there exists $t_{1}$ such that $t_{1}u\in
N^{+}$ and  $\tilde{I}(t_{1}u)<0.$\\
(ii). If $a^+<\mu_p$, $0<\lambda<\min\{\lambda_1,\lambda_{2}\}$ and $B(u)>0$ for $u\in X_0\setminus
\{0\},$  then there exist $t_{1}< t_{2}$
such that $t_{1}u\in N^{+}$, $t_{2}u\in N^{-}$
and $\tilde{I}(t_1u)<0.$
\end{corollary}
%
%
%
{\bf Proof.}(i). From the (\ref{eq2.5c}) and (g1), for
a fixed $u$, we know\,\, $\varphi_u'(0)<0$ and using lemma \ref{lem2.1},
$\lim_{t\rightarrow\infty}\varphi_u'(t)=+\infty$, so by the
intermediate value theorem, there exists\,  $t_{1}>0$  such that
 $\varphi'_u(t_{1})=0$. Now since
 $\varphi_u'(t)<0$ for $0<t<t_{1}$ and  $\varphi'_u(t)>0$  for
 $t_{1}<t$, thus $t_{1}u\in N^{+}$ and
 $\tilde{I}(t_{1}u)<\tilde{I}(0)=0.$
 $\Box$
 
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%
(ii).  As in the proof of (i), we obtain
 $\varphi'_u(0)<0$, $\lim_{t\rightarrow\infty}\varphi'_u(t)=- \infty$ and by using
lemma \ref{lem2.5} we get $\varphi'_u(\tau)>0$ for a suitable $\tau$, so the
intermediate value theorem concludes that there exist
$t_{1},t_{2}$ such that $0<t_{1}<\tau<t_{2}$, $
\varphi'_u(t_{1})=\varphi'_u(t_{2})=0$, $t_{1}u\in
N^{+}$, $t_{2}u\in N^{-}$ and
$\tilde{I}(t_{1}u)<\tilde{I}(0)=0$. 
$\Box$
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

%
%
%           Lemma(2.4)
%
\begin{lemma}\label{lem2.4}
 There exists $\lambda_3>0$ such that if $0<\lambda<\lambda_3$, then $B(u)>0$ provided that $u\in N^-$.
\end{lemma}
%
%
%
{\bf Proof.} Suppose otherwise, that is, $-(q-1)B(u)\geq0$ and by (\ref{eq2.5c})
%
%
%              
%
\begin{equation*}
\varphi_u^{''}(1)=
(p-1)\|u\|^p_{\tilde{A}}-(q-1)B(u)-\lambda\int_{\Omega}g_{u}(x,|u|)u^{2} dx+(r-1)H(u)<0,
\end{equation*}
%
%
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 so using (g1),  $(p-1)\|u\|^p_{\tilde{A}}<\lambda \|\bar{g_2}\|_\infty \tilde{S}_p^p{\|u \|}^p_{\tilde{A}},$
%
%              
%
%
%
which is a contradiction for
$\lambda<\lambda_3:=\frac{p-1}{\|\bar{g_2}\|_\infty S_p^p}.$
$\Box$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%

%
{\bf Proof of Theorem \ref{thm2.6}(i).} Assume $\lambda^*:=\min\{\lambda_0, \lambda_1, \lambda_2\}$, as in lemma \ref{lem2.1}, $\tilde{I}$ is bounded below on
$N$ and so on $N^+$. Let $\{u_{n}\}$ be a
minimizing sequence for $\tilde{I}$ on
$N^+$, i.e., $\lim_{n \to \infty}\tilde{I}(u_{n})= \inf_{u\in
N^+} \tilde{I}(u)=c,$
%
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%
%
%
and by Ekeland's variational principle \cite{Ekeland} we may assume
$\langle\tilde{I'} (u_{n}),u_{n}\rangle \to 0.$

On the other hand, similar to lemma \ref{lem2.1}, $\tilde{I}(u_{n})-\frac{1}{q}\langle\tilde{I'} (u_{n}),u_{n}\rangle\geq C\|u_n\|_{\tilde{A}}-K$, so $\{u_{n}\}$
  is bounded in $X_0$ and
   without loss of generality, we may assume
 that $u_{n}\rightharpoonup u_{1}$ in $X_0$ and
$u_{n}\rightarrow u_{1}$  in $L^{r}(\Omega)$ for $1\leq r<p^{*}$
and $u_{n}(x)\rightarrow u_{1}(x)$, a.e.

 By corollary \ref{cor2.2} for $u_{1}\in X_0\setminus\{0\}$ there exists
$t_{1}$ such that $t_{1}u_{1}\in N^{+}$ and so
$\varphi'_{u_1}(t_1)=0.$ Now we show that $u_{n}\rightarrow
u_{1}$ in $X_0$. Suppose this is false, then $\|u_{1}\|^p_{\tilde{A}}< \liminf_{n\rightarrow \infty}  \|u_{n}\|^p_{\tilde{A}}.$
%
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 %
%
%
  %
%
%
So from (\ref{eq2.4}) and remark \ref{rem1.1},
$\varphi'_{u_n}(t_{1})>\varphi'_{u_1}(t_{1})=0$, for 
sufficiently large $n$. Since $\{u_{n}\}\subseteq
N^{+},$ by considering possible maps it is easy
to see that $\varphi'_{u_n}(t)<0$ for $0<t<1$ and
$\varphi'_{u_n}(1)=0$ for all $n.$ Hence we must have
$t_{1}>1$, but $t_{1}u_{1}\in N^{+}_{\lambda}$ and so
%
%
$\tilde{I}(t_{1}u_{1})<\tilde{I}(u_{1})<\lim_{n \to
\infty}\tilde{I}(u_{n})= \inf_{{u\in N}^{+}}
\tilde{I}(u_{n}),$ which is a contradiction. Therefore
$u_{n}\rightarrow u_{1}$ in $X_0$ and so
%
%
$\tilde{I}(u_{1})= \lim_{n \to \infty}\tilde{I}(u_{n})=
\inf_{u\in N^{+}} \tilde{I}(u).$
%
%
 Thus
$u_{1}$ is a minimizer for $\tilde{I}$ on
$N^{+}$ and by using lemmas \ref{lem2.02} and \ref{lem2.03}, $u_1$ is a nontrivial weak
solution of (\ref{eP2}).
$\Box$
%
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%

%
%
{\bf Proof of Theorem \ref{thm2.6}(ii).} Let $\lambda^{**}:=\min\{\lambda_0, \lambda_1, \lambda_2, \lambda_3\}$, then by lemma \ref{lem2.5} for all
$u\in N^{-}$  we have $\tilde{I}(u)\geq \tilde{I}(t_{\max} u)>\epsilon>0$  i.e., $\inf_{u\in N^{-}} \tilde{I}(u)\geq0.$
%
%
 Hence there exists a minimizing sequence $\{u_{n}\}\subseteq
N^{-}$ such that $\lim_{n \to \infty}\tilde{I}(u_{n})= \inf_{u\in
N^{-}} \tilde{I}(u)\geq0$. Now 
%
%
%
%
similarly as in the proof of the pervious theorem we find that,
$\{u_{n}\}$ is bounded in $X_0$,
$u_{n}\rightharpoonup u_{2}$ in $X_0$ and
$u_{n}\rightarrow u_{2}$ in $L^{r}(\Omega)$, $1<r<p^{*}.$  Since $u_n \in N^-$ so by lemma \ref{lem2.4}, $B(u_n)>0$ and $B(u_2)\geq0$. 
%
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%
%
We claim that $B(u_{2})>0$. Suppose this is false, thus
$(p-1)\|u_2\|^p_{\tilde{A}}<\lambda \|\bar{g_2}\|_\infty \tilde{S}_p^p{\|u_2\|}^p_{\tilde{A}},$
%
%
%
 %
  which gives a
contradiction for $\lambda<\lambda_3$. 
So by corollary \ref{cor2.2} there exists $t_{2}>0$ such that $
t_{2}u_{2}\in N^{-}$. We claim that
$u_{n}\rightarrow u_{2}$ in $X_0$; if it is supposed that
this is false, so $\|u_2\|^p_{\tilde{A}}<\liminf_{n \rightarrow \infty}\|u_n\|^p_{\tilde{A}}$.
%
%
 %
 %
 But $u_{n}\in N^{-}$ and so
$\tilde{I}(u_{n})\geq \tilde{I}(tu_{n})$ for all $t\geq0$.
Therefore, using remark \ref{rem1.1} we
get
%
%
%
\begin{equation*}\begin{split}
 \tilde{I}(t_{2}u_{2})&=
\frac{1}{p}t^{p}_{2}\|u_2\|^p_{\tilde{A}}-\frac {1}{q}t_{2}^{q}B(u_{2})-
\lambda\int_{\Omega}G(x,t_{2}|u_{2}|)dx+\frac{t_2^r}{r}H(u_2)\\
 & < \lim_{n \rightarrow \infty}
\big(\frac{1}{p}t^{p}_{2}\|u_n\|^p_{\tilde{A}}-\frac {1}{q}t_{2}^{q}B(u_{n})-
\lambda\int_{\Omega}G(x,t_{2}|u_{n}|)dx+\frac{t_2^r}{r}H(u_n)\big)\\
& = \lim_{n \rightarrow \infty}\tilde{I}(t_{2}u_{n})\leq
\lim_{n \rightarrow \infty} \tilde{I}(u_{n})= \inf_{u\in
N^{-}} \tilde{I}(u),
\end{split}\end{equation*}
which is a contradiction, hence by lemmas \ref{lem2.02} and \ref{lem2.03}, $u_2$ is a nontrivial weak
solution of (\ref{eP2}) which belongs to $N^-$. Moreover $N^+\cap N^-=\emptyset$, so $u_1$ and $u_2$ are distinct.
$\Box$
%
%
% 
%
%%%%%%%%%%%%%%%%%%%
%


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\end{thebibliography}


\end{center}



{\small

\noindent{\bf Farajollah Mohammadi Yaghoobi}

\noindent Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent Hamedan Branch, Islamic Azad University

\noindent Hamedan, Iran

\noindent E-mail: yaghoobi@iauh.ac.ir}\\

{\small
\noindent{\bf  Jamileh Shamshiri  }

\noindent  Department of Accounting

\noindent Assistant Professor of Mathematics

\noindent Tabaran Institute of Higher Education


\noindent Mashhad, Iran

\noindent E-mail: jamileshamshiri@gmail.com}\\



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