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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 97, 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 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/97\hfil Critical elliptic problems with fractional Laplacian]
{Multiple solutions for critical elliptic problems with fractional Laplacian}

\author[G. Lin, X. Zheng \hfil EJDE-2016/97\hfilneg]
{Guowei Lin, Xiongjun Zheng}

\address{Guowei Lin \newline
Department of Mathematics,
Jiangxi Normal University,
Nanchang, Jiangxi 330022, China}
\email{lgw2008@sina.cn}

\address{Xiongjun Zheng \newline
Department of Mathematics,
Jiangxi Normal University,
Nanchang, Jiangxi 330022, China}
\email{xjzh1985@126.com}

\thanks{Submitted January 11, 2016. Published April 14, 2016.}
\subjclass[2010]{35J60, 35J61, 35J70, 35J99}
\keywords{Fractional Laplacian, critical exponent, multiple solutions, category}

\begin{abstract}
 This article is devoted to the study of the  nonlocal fractional equation
 involving critical nonlinearities
  \begin{gather*}
 (-\Delta)^{\alpha/2} u=\lambda u+|u|^{2^{\ast}_{\alpha}-2}u \quad 
 \text{in } \Omega,\\
 u=0 \quad \text{on } \partial \Omega,
 \end{gather*}
 where  $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $N \geq 2\alpha$,
 $\alpha\in(0,2)$, $ \lambda\in(0,\lambda_{1})$ and
 $2^*_{\alpha}=\frac{2N}{N-\alpha}$ is critical exponent.
 We show the existence of at least $\operatorname{cat}_{\Omega}(\Omega) $
 nontrivial solutions for this problem.
\end{abstract}

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

\section{Introduction}

This article concerns the critical elliptic problem with the fractional Laplacian
\begin{equation}\label{e1.1}
\begin{gathered}
(-\Delta)^{\alpha/2} u=\lambda u+|u|^{2^{\ast}_{\alpha}-2}u \quad
\text{in } \Omega,\\
u=0 \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$ with
 $N> \alpha$, $\alpha \in (0,2)$ is fixed and  
$2^{\ast}_{\alpha}=\frac{2N}{N-\alpha}$ is the critical Sobolev exponent.

In a bounded domain $\Omega\subset \mathbb{R}^N$, the operator 
$(-\Delta)^{\alpha/2}$ can be defined as in \cite{BCPa,CT} as follows.
Let $\{(\lambda_k,\varphi_k)\}^\infty_{k=1}$ be the eigenvalues and 
corresponding eigenfunctions of the
Laplacian $-\Delta$ in $\Omega$ with zero Dirichlet boundary values 
on $\partial\Omega$ normalized by $\|\varphi_k\|_{L^2(\Omega)} = 1$, i.e.
\[
-\Delta \varphi_k = \lambda_k \varphi_k\quad{\rm in}\ \Omega;\quad 
\varphi_k = 0\quad{\rm on}\ \partial\Omega.
\]
We define the space $H^{\alpha/2}_0(\Omega)$ by
\[
H^{\alpha/2}_0(\Omega)=\{u=\sum_{k=1}^\infty u_k\varphi_k\in L^2(\Omega): \sum_{k=1}^\infty u_k^2\lambda_k^{\frac \alpha2}<\infty\},
\]
which is equipped with the norm
\[
\|u\|_{H^{\alpha/2}_0(\Omega)} =\Big(\sum_{k=1}^\infty 
 u_k^2\lambda_k^{\frac \alpha2}\Big)^{\frac 12}.
\]
For $u\in H^{\alpha/2}_0(\Omega)$, the fractional Laplacian $(-\Delta)^{\alpha/2}$ is defined by
\[
(-\Delta)^{\alpha/2}u = \sum_{k=1}^\infty u_k\lambda_k^{\alpha/2}\varphi_k.
\]

Problem \eqref{e1.1} is the Br\'{e}zis-Nirenberg type problem
with the fractional Laplacian.  Br\'{e}zis and Nirenberg \cite{BN}
considered the existence of positive solutions for problem \eqref{e1.1}
with $\alpha = 2$. Such a problem involves
 the critical Sobolev exponent $2^* = \frac {2N}{N-2}$ for $N\geq 3$, and
 it is well known that the Sobolev embedding
 $H^1_0(\Omega)\hookrightarrow L^{2^*}(\Omega)$ is not compact even
 if $\Omega$ is bounded. Hence, the associated functional of problem \eqref{e1.1}
does not satisfy the Palais-Smale condition, and critical point theory cannot 
be applied directly to find solutions of the problem. However, 
it is found in \cite{BN} that the functional satisfies the  $(PS)_c$ condition 
for $c\in (0, \frac 1N S^{N/2})$, where $S$ is the best Sobolev
constant and $\frac 1N S^{N/2}$ is the least level at which the
Palais-Smale condition fails. So a positive solution can be found if the 
mountain pass value corresponding to problem \eqref{e1.1} is strictly
less than $\frac 1N S^{N/2}$.

Problems with the fractional Laplacian  have been extensively
studied, see for example \cite{BCP, BCPa, CabS, CT, CDDS,
CSS, CS, T, TX} and the references therein. In
particular, the Br\'{e}zis-Nirenberg type problem was discussed in
\cite{T} for the special case $\alpha = \frac 12$,  and in
\cite{BCP} for the general case, $0<\alpha<2$, where existence of
one positive solution was proved.  To use the idea in \cite{BN} to
prove the existence of one positive solution for the fractional
Laplacian, the authors in \cite{BCP, T} used the following results
in \cite{CS} (see also \cite{BCPa}):  for any $u\in H^\alpha_0(\Omega)$, 
the solution $v\in H^1_{0,L}(\mathcal{C}_\Omega)$ of the problem
\begin{equation}\label{e1.2}
\begin{gathered}
-\operatorname{div}(y^{1-\alpha}\nabla v) = 0, \quad \text{in }
 \mathcal{C}_{\Omega}=\Omega\times(0,\infty),\\
v=0,\quad \text{on } \partial_L\mathcal{C}_{\Omega}=\partial\Omega\times(0,\infty),\\
v = u , \quad\text{on }  \Omega\times\{0\},
 \end{gathered}
 \end{equation}
satisfies  
$$
-\lim_{y\to 0^+}k_\alpha y^{1-\alpha}\frac {\partial v}{\partial y} 
= (-\Delta)^{\alpha}u,
$$
where  we use  $(x,y)= (x_1,\dots,x_N,y)\in \mathbb{R}^{N+1}$, and
\begin{equation}\label{e1.3}
H^1_{0,L}(\mathcal{C}_\Omega) 
= \big\{w\in L^2(\mathcal{C}_\Omega): w =
0 \text{ on } \partial_L\mathcal{C}_\Omega,
\int_{\mathcal{C}_\Omega}y^{1-\alpha}|\nabla w|^2\,dx\,dy<\infty\big\}.
\end{equation}
 Therefore,  the nonlocal problem \eqref{e1.1} can be reformulated
as the local problem
\begin{equation}\label{e1.4}
\begin{gathered}
-\operatorname{div}(y^{1-\alpha}\nabla w) = 0,  \quad
 \text{in }\mathcal{C}_{\Omega},\\
v=0, \quad \text{on } \partial_L\mathcal{C}_{\Omega},\\
  \lim_{y\to 0^+}y^{1-\alpha}\frac {\partial w}{\partial \nu} 
= |w(x,0)|^{2^*_\alpha-2}w(x,0)+ \lambda w(x,0),  \quad \text{on }
\Omega\times\{0\},
 \end{gathered}
 \end{equation}
where $\frac {\partial }{\partial \nu}$ is the outward normal
derivative of $\partial \mathcal{C}_{\Omega}$. Hence, critical
points of the functional
\begin{equation}\label{e1.5}
\begin{aligned}
J(w) &= \frac 12 \int_{\mathcal{C}_\Omega}y^{1-\alpha}|\nabla w|^2\,dx\,dy 
-\frac 1{2^*_\alpha}\int_{\Omega\times\{0\}} |w(x,0)|^{2^*_\alpha}\,dx \\
 &\quad -\frac \lambda 2\int_{\Omega\times\{0\}} |w(x,0)|^2\,dx
\end{aligned}
\end{equation}
defined on $H^1_{0,L}(\mathcal{C}_\Omega)$ correspond to solutions
of \eqref{e1.4}, and the trace $u = tr\, w$ of $w$ is a solution 
of \eqref{e1.1}. A critical point of the
functional $J(u)$ at the mountain pass level was found in \cite{BCP, T}.
 On the other hand, it can be shown by using the Pohozaev type identity
 that the problem
\begin{gather*}
(-\Delta)^{\alpha/2} u= |u|^{p-1}u \quad \text{ in } \Omega,\\
u=0\quad  \text{on }  \partial\Omega
\end{gather*}
has no nontrivial solution if $p+1\geq \frac{2N}{N-\alpha}$ and $\Omega$ 
is star-shaped, see for example \cite{BCPa} and \cite{T}.

It is well-known that if $\Omega$ has a rich topology,  \eqref{e1.1}
with $\alpha = 2,\, \lambda = 0$ has
a solution, see \cite{BC, CP, W} etc. In this paper, we assume 
$0<\lambda<\lambda_1$, where $\lambda_1$ is the first eigenvalue 
of the fractional Laplacian $(-\Delta)^{\alpha/2}$. We investigate 
the existence of multiple solutions of problem \eqref{e1.1}.
Let $A$ be a closed subset of a topology space $X$. The category of $A$ 
is the least integer $n$ such that there exist $n$ closed subsets 
$A_1,\dots, A_n$ of $X$ satisfying $A = \cup_{j=1}^n A_j$ and 
$A_1,\dots, A_n$ are contractible in $X$. Our main result is as follows.

\begin{theorem}\label{thm:1.1}
If $\Omega$ is a smooth bounded domain of $\mathbb{R^{N}}$, $N\geq 4,\,0<\alpha<2$ 
and $0<\lambda<\lambda_{1}$, problem \eqref{e1.4} has at least
 $\operatorname{cat}_{\Omega}(\Omega)$ nontrivial solutions. 
Equivalently,  \eqref{e1.1}
possesses at least $\operatorname{cat}_{\Omega}(\Omega)$ positive solutions.
\end{theorem}


We say that $w \in H^1_{0,L}(\mathcal{C}_{\Omega})$ is a solution to
\eqref{e1.4} if for every function
$ \varphi \in H^1_{0,L}(\mathcal{C}_{\Omega})$, we have
\begin{equation}\label{e1.6}
k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}\langle\nabla w ,
\nabla \varphi \rangle\, dx\,dy=\int_{\Omega}
(\lambda w+w^{\frac{N+\alpha}{N-\alpha}})\varphi\,dx.
\end{equation}
We will find solutions of $J$ at energy levels below a value related
 to the best Sobolev constant $S_{\alpha,N}$, where 
\begin{equation}\label{e1.7}
S_{\alpha,N}=\inf_{w\in H^1_{0,L}({\mathcal{C}_{\Omega}}),w\neq 0}
\frac{k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}|\nabla w|^2\,dx\,dy}
{(\int_{\Omega}|w(x,0)|^{2^*_{\alpha}}\,dx)^{2/(2^*_{\alpha})}},
\end{equation}
 which is not achieved in any bounded domain and is indeed achieved in 
the case $\Omega=\mathbb{R}^{N+1}_{+}$. We know from \cite{BCP} that 
the trace $u_{\epsilon}(x)=w_{\epsilon}(x,0)$ of the family of minimizers 
$w_{\epsilon}$ of $S_{\alpha,N}$ takes the form
\begin{equation}\label{e1.8}
u(x)=u_{\epsilon}(x)=\frac{\epsilon^{\frac{N-\alpha}{2}}}{(|x|^2+\epsilon^2)
^{\frac{N-\alpha}{2}}},
\end{equation}
with $\epsilon>0$. Using this property, we are able to find critical values 
of $J$ in a right range.

  In section 2, we prove the $(PS)_c$ condition and the main result is
 shown in section 3.


 \section{Palais-Smale condition}

In this section, we show that the functional $J(w)$ satisfies $(PS)_c$ condition 
for $c$ in certain interval.
By a $(PS)_c$ condition for the functional $J(w)$ we mean that a sequence 
$\{w_n\}\subset H^1_{0,L}(\mathcal{C}_{\Omega})$ such that
$J(w_n)\to c,\, J'(w_n)\to 0$ contains a convergent subsequence.

Define on the space $H^1_{0,L}(\mathcal{C}_{\Omega})$ the functionals
\begin{gather*}
\psi(w)=\int_{\Omega}(w^{+}(x,0))^{2^*_{\alpha}}\,dx,\\
\varphi_{\lambda}(w)=k_{\alpha}\int_{C_{\Omega}}y^{1-\alpha}
|\nabla w|^2dx\,dy-\lambda \int_{\Omega}|w(x,0)|^2dx.
\end{gather*}
We may verify as in \cite{W} that on the manifold
$$
V=\{w\in  H^1_{0,L}(\mathcal{C}_{\Omega}):\psi (w)=1\},
$$
$\psi'(w)\neq 0$ for every $w\in V$. Hence, the tangent space of 
$V$ at $v$ is given by
$$
T_{v}V :=\{w\in  H^1_{0,L}(\mathcal{C}_{\Omega}): \langle \psi'(v) , w\rangle = 0 \},
$$
and the norm of the derivative of $ \varphi_{\lambda}(w)$ at $v$ restricted to
  $V$ is defined by
$$
 \|\varphi'_{\lambda}(v)\|_{\ast}=\sup_{w\in T_{v}V ,\|w\|=1}
|\langle\varphi'_{\lambda}(v),w\rangle|.
$$
It is well known that
$$ 
\|\varphi_{\lambda}'(w)\|_{\ast}=\min_{\mu\in\mathbb{R}}
 \| \varphi'_{\lambda}(w)-\mu\psi' (w) \|.
$$
A critical point $v\in V$ of  $\varphi_{\lambda}$ is a point such that  
$\|\varphi'_{\lambda}(v)\|_{\ast} = 0$.

Since $\lambda_1$ is the first eigenvalue of  the fractional Laplacian 
$(-\Delta)^{\alpha/2}$, it can be characterized as
$$
\lambda_{1}=\inf_{w\in H^1_{0,L}({C_{\Omega}}),w\neq 0}
\frac{k_{\alpha}\int_{C_{\Omega}}y^{1-\alpha}|\nabla w|^2\,dx\,dy}
{\int_{\Omega}|w(x,0)|^2\,dx}.
$$
If $0<\lambda<\lambda_{1}$, we see that
$$
\|w\|_{1} :=\Big(k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}|\nabla w|^2
\,dx\,dy-\lambda \int_{\Omega} w^2(x,0)\,dx\Big)^{1/2}
$$
is an equivalent norm on $H^1_{0,L}(\mathcal{C}_{\Omega})$.

\begin{lemma}\label{lm:2.1}
 Any sequence $\{v_{n}\}\subset H^1_{0,L}(\mathcal{C}_{\Omega})$ such that
 $$ 
d :=\sup_{n}J(v_{n})<C^{\ast}:=\frac{\alpha}{2N}S_{\alpha,N}^{\frac{N}{\alpha}},
\quad J'(v_{n})\to 0 \quad \text{in }H^{-1}_{0,L}(\mathcal{C}_{\Omega})
$$
contains a convergent subsequence.
\end{lemma}

\begin{proof} 
It is easy to show from the assumptions that
\begin{align*}
d+1+\|v_{n}\|_{1}
&\geq J(v_{n})-\frac{1}{2^{\ast}_{\alpha}}\langle J'(v_{n}), v_{n}\rangle\\
&=(\frac{1}{2}-\frac{1}{2^{\ast}_{\alpha}})
 \Big(\int_{\mathcal{C}_{\Omega}}k_{\alpha}y^{1-\alpha}|\nabla v_{n}|^2\,dx\,dy
 -\lambda\int_{\Omega}|v_{n}|^2\,dx\Big)\\
&=(\frac{1}{2}-\frac{1}{2^{\ast}_{\alpha}})\|v_{n}\|_{1}^2;
\end{align*}
that is, $ \|v_{n}\|_{1}$ is bounded. We may assume that
\begin{gather*}
v_{n}(x,y) \rightharpoonup v(x,y)\quad \text{in }   H^1_{0,L}(\mathcal{C}_{\Omega}),\\
v_{n}(x, 0) \to  v(x,0)\quad \text{in }   L ^2(\Omega),\\
v_{n}(x, 0) \to  v(x,0)\quad\text{ a.e. in } \Omega .
\end{gather*}
Therefore, for every $ \varphi\in H^1_{0,L}(\mathcal{C}_{\Omega})$,
$$
\langle J'(v_{n}),\varphi\rangle\to \langle J'(v ),\varphi\rangle = 0
$$
as $n\to +\infty $. We also have that $J(v)\geq 0$. By Br\'{e}zis-Lieb's lemma,
$$
J(v)+\frac 12\|v_{n}-v\|_{1}^2-\frac 1{2^{\ast}_{\alpha}}
\int_{\Omega}(v_n-v)_{+}^{2^{\ast}_{\alpha}}\,dx=J(v_{n})+o(1)
= C +o(1).
$$
Since $\langle J'(v_{n}),v_{n}\rangle \to 0 $, we obtain
\begin{align*}
&\|v_{n}-v\|_{1}^2-2^{\ast}_{\alpha}
\int_{\Omega}(v_n-v)_{+}^{2^{\ast}_{\alpha}}\,dx\\
&=\|v_{n}\|_{1}^2-\|v\|_{1}^2-2^{\ast}_{\alpha}
 \int_{\Omega}\big((v_n)_{+}^{2^{\ast}_{\alpha}}
- v_{+}^{2^{\ast}_{\alpha}}\big)\,dx+o(1)\\
&=-\|v\|_{1}^2 + 2^{\ast}_{\alpha}\int_{\Omega}v_{+}^{2^{\ast}_{\alpha}}\,dx\\
&=-\langle J'(v),v \rangle 
=0.
\end{align*} 
Hence, there exist a constant $b$ such that
$$
\|v_{n}-v\|_{1}^2\to b,\quad 2^{\ast}_{\alpha}
\int_{\Omega} (v_n)_{+}^{2^{\ast}_{\alpha}}\,dx \to b,\quad \text{as } 
 n\to +\infty.
$$
It follows by $v_n\to v$ in $L^2(\Omega)$ that
$$
\int_{\mathcal{C}_{\Omega}}k_{\alpha}y^{1-\alpha}|\nabla (v_n-v)|^2\,dx\,dy \to b.
$$
The trace inequality
$$
\int_{\mathcal{C}_{\Omega}}k_{\alpha}y^{1-\alpha}|\nabla(v_n-v)|^2\,dx\,dy
\geq S_{\alpha,N}\|(v_n-v)(x,0)\|^2_{L^{2^{\ast}_{\alpha}}(\Omega)}
$$
implies $b\geq S_{\alpha,N}b^{\frac{2}{2^{\ast}_{\alpha}}}$. 
Hence, either $b=0$  or $b\geq S_{\alpha,N} ^{\frac{N}{\alpha}}$.

If $b=0$, then $v_{n}\to v$ in $H_{0,L}^1(\mathcal{C}_{\Omega})$, and
the proof is complete.
If $b\geq S_{\alpha,N} ^{\frac{N}{\alpha}}$, we deduce that
\begin{align*}
C^{\ast}
&=\frac{\alpha}{2N}S^{\alpha/N}_{\alpha,N}\\
&\leq(\frac{1}{2}-\frac{1}{2^{\ast}_{\alpha}})b\\
&=(\frac{1}{2}-\frac{1}{2^{\ast}_{\alpha}})\|v_n - v\|_{1}^2+o(1)\\
&\leq J(v)+\frac12\|v_n - v\|_{1}^2-\frac1{2^{\ast}_{\alpha}}\|v_n - v\|_{1}^2+o(1)\\
&=J(v)+\frac 12\|v_n - v\|_{1}^2-\frac 1{2^{\ast}_{\alpha}}\int_{\Omega}(v_n-v)_{+}^{2^{\ast}_{\alpha}}+o(1)\\
&=C\leq d <C^{\ast},
\end{align*}
which is a contradiction.
\end{proof}

Alternatively, we have the following result.

\begin{lemma}\label{lm:2.2}
Every sequence $\{w_{n}\} \in V $ satisfying   
$\varphi_{\lambda}(w_{n})\to c< S_{\alpha,N}$ and
$\|\varphi'_{\lambda}(w_{n})\|_{\ast}\to 0$, as $n\to +\infty$, 
 contains a convergent subsequence.
\end{lemma}

\begin{proof} Since
$$
\|\varphi'_{\lambda}(w_{n})\|_{\ast}= \min_{\mu\in \mathbb{R}}
\|\varphi'_{\lambda}(w_{n})-\mu\psi'(w_{n})\|,
$$
there exists a sequence $\{\alpha_{n}\}\subset\mathbb{R}$ such that 
$\|\varphi'_{\lambda}(w_{n})-\alpha_{n}\psi'(w_{n})\| \to 0 $. 
It follows that for every $ h\in H_{0,L}^1(\mathcal{C}_{\Omega})$,
\begin{equation}\label{e2.1}
 k_{\alpha}\int_{C_{\Omega}}y^{1-\alpha}\nabla w_{n}\nabla h\,dx\,dy
 -\lambda\int_{\Omega}w_{n}h\,dx -\mu_{n}
\int_{\Omega}(w_{n}^{+})^{2_{\alpha}^{\ast}-1}h\,dx \to 0,
\end{equation}
where $\mu_{n}=\frac{\alpha_{n}2_{\alpha}^{\ast}}{2}$.
 Choosing $h=w_{n}$ in \eqref{e2.1} and using the fact $\psi(w_{n}^{+})=1$,
we obtain
$$
\varphi_{\lambda}(w_{n})-\mu_{n}=k_{\alpha}
\int_{C_{\Omega}}y^{1-\alpha}|\nabla w_{n}|^2
-\lambda\int_{\Omega}|w_{n}|^2-\mu_{n}
\int_{\Omega}(w_{n}^{+})^{2^{\ast}_{\alpha}}\to 0.
$$
Whence by $ \varphi_{\lambda}(w_{n})\to c,$ $\mu_{n}\to c$ as 
$n \to +\infty$. Setting $v_{n}:=\mu_{n}^{\frac{N-\alpha}{2\alpha}}w_{n}$, we obtain
\begin{align*}
J(v_{n} )
& =\frac{1}{2}\mu_{n}^{\frac{N-\alpha}{\alpha}}
 \Big(\int_{C_{\Omega}}k_{\alpha}y^{1-\alpha}|\nabla w_{n}|^2\,dx\,dy
 -\lambda\int_{\Omega}|w_{n}|^2\,dx\Big)
 -\frac{1}{2^{\ast}_{\alpha}}\mu_{n}^{\frac{N}{\alpha}}
 \int_{\Omega}(w_{n}^{+})^{2^{\ast}_{\alpha}}\\
&=\frac{1}{2}\mu_{n}^{\frac{N-\alpha}{\alpha}}\varphi_{\lambda}(w_{n})
 -\frac{N-\alpha}{2N}\mu_{n}^{\frac{N}{\alpha}}.
\end{align*}
Thus, 
$$
J(v_{n})\to \frac{\alpha}{2N}c^{\frac{N}{\alpha}}
<\frac{\alpha}{2N}S^{\frac{N}{\alpha}}.
$$
In the same way, for every $ h\in H_{0,L}^1(\mathcal{C}_{\Omega})$, 
by \eqref{e2.1},
\begin{align*}
&\langle J'(v_{n} ),h\rangle \\
& =\mu_{n}^{\frac{N-\alpha}{2\alpha}}\Big(k_{\alpha}
\int_{\mathcal{C}_{\Omega}}y^{1-\alpha} \nabla w_{n}\nabla h\,dx\,dy 
-\lambda\int_{\Omega}w_{n}h-\mu_{n}\int_{\Omega}(w_{n}^{+})^{2^{\ast}_{\alpha}-1}h
\,dx\Big)\to 0.
\end{align*}
Now, the assertion follows by Lemma \ref{lm:2.1}.
\end{proof}


Let us define 
$$
 Q_\lambda =\inf_{w\in V}\varphi_{\lambda}(w)=\inf_{w\in V}
\big\{k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}|\nabla w|^2dx\,dy
-\lambda \int_{\Omega}|w(x,0)|^2\,dx\big\}.
$$
Denote by $\eta_0(t)\in C^{\infty}(\mathbb{R_{+}})$ a cut-off function, 
which is non-increasing and satisfies
$$
\eta_0(t)=\begin{cases}
1 & \text{if } 0\leq t\leq\frac{1}{2},\\
0 &\text{if } t\geq1.
\end{cases}
$$
Assume $0\in\Omega$, for fixed $\rho>0$ small enough such that 
$\overline B_{\rho} \subseteq C_{\Omega}$, we define the function
$\eta(x,y)=\eta_{\rho}(x,y)=\eta_0(\frac  {|(x,y)|}{\rho})$.
Then $\eta w_{\epsilon}\in H_{0,L}^1(C_{\Omega})$.
It is standard to establish the following estimates, see \cite{BCP} for details.

\begin{lemma}\label{lm:2.3}
 The family $\{\eta w_{\epsilon}\}\subset H_{0,L}^1(\mathcal{C}_{\Omega})$ 
and its trace on ${y=0}$ satisfy
 \begin{equation}\label{e2.2}
\|\eta w_{\epsilon}\|^2=\|w_{\epsilon}\|^2+O(\epsilon^{N-\alpha}),
\end{equation}
If $N>2\alpha$,
\begin{equation}\label{e2.3}
\|\eta w_{\epsilon}\|^2_{L^2(\Omega)}=C\epsilon^{\alpha}+O(\epsilon^{N-\alpha}),
\end{equation}
If $N=2\alpha$,
\begin{equation}\label{e2.4}
\|\eta w_{\epsilon}\|^2_{L^2(\Omega)}=C\epsilon^{\alpha}
\log (\frac{1}{\epsilon})+O(\epsilon^{\alpha})
\end{equation}
for $\epsilon>0$ small enough and some $C>0$.
\end{lemma}

\begin{lemma}\label{lm:2.4}
Assume $N\geq2\alpha$, $0< \lambda<\lambda_{1}$, then
\begin{equation}\label{e2.5}
Q_\lambda =\inf_{w\in V}\varphi_{\lambda}(w)<S_{\alpha,N}.
\end{equation}
Moreover, there exists $u\in V$ such that $\varphi_{\lambda}(u)=Q_\lambda$.
 \end{lemma}

\begin{proof}
We first show that \eqref{e2.5} holds  if $N\geq2\alpha$ and
$ 0<\lambda<\lambda_{1}$. Since
$$
\int_{|x|>\frac{\rho}{2}}|u_{\epsilon}|^{2^{\ast}_{\alpha}}\,dx
=\int_{\{|x|\geq\frac{\rho}{2}\}}\frac{\epsilon^{N}}{(|x|^2+\epsilon^2)^{N}}\,dx
\leq\frac{N2^{N}}{\rho^{N}}\epsilon^{N},
$$
we have
\begin{align*}
\int_{\Omega}|\eta u_{\epsilon}|^{2^{\ast}_{\alpha}}\,dx
&\geq\int_{\{|x|\leq\frac{\rho}{2}\}}|u_{\epsilon}|^{2^*_{\alpha}}dx
=\|u_{\epsilon}\|^{2^*_{\alpha}}_{L^{2^*_{\alpha}}(\Omega)}
-\int_{\{|x|\geq\frac{\rho}{2}\}}|u_{\epsilon}|^{2^*_{\alpha}}dx\\
&\geq\|u_{\epsilon}\|^{2^{\ast}_{\alpha}}_{L^{2^*_{\alpha}}(\Omega)}+O(\epsilon^{N}).
\end{align*}
By  Lemma \ref{lm:2.3},  for $N>2\alpha$, we have
\begin{align*}
&\frac{k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}
 |\nabla( \eta w_{\epsilon})|^2\,dx\,dy
 -\lambda\int_{\Omega}| \eta u_{\epsilon}|^2\,dx}
 {(\int_{\Omega}|\eta u_{\epsilon}|^{2^*_{\alpha}}\,dx)^{\frac{2}{2^*_{\alpha}}}}\\
&\leq\frac{k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}
 |\nabla w_{\epsilon}|^2\,dx\,dy
 -\lambda C\epsilon^{\alpha}+O(\epsilon^{N-\alpha})}
{\|u_{\epsilon}\|^2_{L^{2^*_{\alpha}}(\Omega)}+O(\epsilon^{N})}\\
&\leq S_{\alpha,N}-\frac{\lambda C\epsilon^{\alpha}}
 {\|u_{\epsilon}\|^2_{L^{2^*_{\alpha}(\Omega)}}}+O(\epsilon^{N-\alpha})
<S_{\alpha,N}.
\end{align*}
Similarly, for $N=2\alpha$, we find for $\epsilon$ small enough such that
 $$
Q_\lambda\leq S_{\alpha,N}-\frac{\lambda C\epsilon^{\alpha}
\log(\frac{1}{\epsilon})}{\|u_{\epsilon}\|^2_{L^{2^*_{\alpha}
(\Omega)}}}+O(\epsilon^{\alpha})
 <S_{\alpha,N}.
$$
Consequently, inequality \eqref{e2.5} holds.

Next, we show that $Q_\lambda$ is achieved if $0<\lambda<\lambda_{1}$. 
Obviously, $Q_\lambda>0$.  Now, let  
$\{w_{n}\}\subset H^1_{0,L}(\mathcal{C}_{\Omega})$ be a minimizing sequence of  
$Q_\lambda>0$ such that $w_{n}\geq0$ and 
$\|w_{n}(x,0)\|_{L^{2^*_{\alpha}}(\Omega)}=1$. The boundedness of
 $\{w_{n}\}$ implies that
\begin{gather*}
w_{n}(x,y)  \rightharpoonup w(x,y)\quad \text{in } 
   H^1_{0,L}(\mathcal{C}_{\Omega}),\\
w_{n}(x, 0) \to w(x,0)\quad \text{in }   L^{q}(\Omega),\\
w_{n}(x, 0)\to w(x,0)\quad \text{a.e. in} \quad \Omega,
\end{gather*}
where $1\leq q\leq 2^*_{\alpha}$. Since
 $$
\|w_{n}\|^2 = \|w_{n}-w\|^2+\|w\|^2+o(1),
$$
 by the Brezis-Lieb Lemma,
\begin{align*}
&\|w_{n}\|^2-\lambda\|w_{n}(x,0)\|^2_{L^2(\Omega)}\\
&=\|w_{n}-w\|^2+\|w\|^2-\lambda\|w_{n}(x,0)\|^2_{L^2(\Omega)}+o(1)\\
&\geq S_{\alpha,N}\|w_{n}(x,0)-w(x,0)\|^2_{L^{2^*_{\alpha}}(\Omega)}
 + Q_\lambda\|w(x,0)\|^2_{L^{2^{\ast}_{\alpha}}(\Omega)}+o(1)\\
&\geq(S_{\alpha,N}-Q_\lambda)\|w_{n}(x,0)-w(x,0)
 \|^{2^{\ast}_{\alpha}}_{L^{2^*_{\alpha}}(\Omega)}
 + Q_\lambda\|w_{n}(x,0)\|^{2^{\ast}_{\alpha}}_{L^{2^{\ast}_{\alpha}}(\Omega)}+o(1)\\
&=(S_{\alpha,N}-Q_\lambda)\|w_{n}(x,0)-w(x,0)
 \|^{2^*_{\alpha}}_{L^{2^*_{\alpha}}(\Omega)}+ Q_\lambda+o(1).
\end{align*}
Hence, we obtain
$$
 o(1)+Q_\lambda\geq(S_{\alpha,N}
-Q_\lambda)\|w_{n}(x,0)-w(x,0)\|^{2^{\ast}_{\alpha}}_{L^{2^*_{\alpha}}(\Omega)}
+Q_\lambda +o(1).
$$
The  $S_{\alpha,N}>Q_\lambda$ implies
$w_{n}(x,0)\to w(x,0)$  in  $ L^{2^{\ast}_{\alpha}}(\Omega)$  and 
$ \|w(x,0)\|_{L^{2^{\ast}_{\alpha}}(\Omega)}=1$. This yields
$$
Q_\lambda \leq\|w\|^2-\lambda\|w(x,0)\|^2_{L^2(\Omega)}
 \leq\lim_{n\to+\infty}(\|w_{n}\|^2-\lambda\|w_{n}(x,0)\|^2_{L^2(\Omega)})
\leq Q_\lambda;
$$
that is, $w$ is a minimizer for $Q_\lambda$.
\end{proof}


 \section{Proof of main theorem}

Taking into account the concentration-compactness principle in \cite{L},
 we may derive the following result, its proof can be found in  \cite{BCP}.

\begin{lemma}\label{lm:3.1}
Suppose ${w_{n}}\rightharpoonup w$  in $H^1_{0,L}(\mathcal{C}_{\Omega})$,
 and the sequence $\{y^{1-\alpha}|\nabla w_{n}|^2\}$ is tight, i.e. for
any $\eta>0$ there exists $\rho_0>0$ such that for all $n$,
\[
\int_{\{y>\rho_0\}}\int_\Omega y^{1-\alpha}|\nabla w_{n}|^2\,dx\,dy<\eta.
\]
Let $u_{n}=t_{r}w_{n}$ and $u=t_{r}w$ and let $\mu$, $\nu$  be two non 
negative measures such that
   \begin{equation}\label{e3.1}
 y^{1-\alpha}|\nabla w_{n}|^2\to \mu \quad \text{and} \quad
|u_{n}|^{2^{\ast}_{\alpha}}\to \nu
\end{equation}
in the sense of measures as $n\to\infty $. Then, there exist an at most 
countable set $I$ and points ${x_{i}}\in \Omega$ with $i\in I$ such that
\begin{itemize}
\item[(1)] $\nu =|u|^{2^{\ast}_{\alpha}}+\Sigma_ {k\in I} \nu_{k}\delta_{x_{k}}$,
 $\nu_{k} >0$,

\item[(2)] $\mu =y^{1-\alpha}|\nabla w|^2+\Sigma_ {k\in I} \mu_{k}\delta_{x_{k}}$,
 $\mu_{k}>0$,

\item[(3)] $ \mu_{k} \geq S_{\alpha,N} \nu_{k}^{\frac{2}{2^{\ast}_{\alpha}}}$.
\end{itemize}
\end{lemma}


On the manifold $V$, we define the mapping $\beta: V\to \mathbb{R}^N$ by
 $$
\beta (w):=\int_{\Omega}x(w^{+}(x,0))^{2^*_{\alpha}}\,dx,
$$
which has the following properties.

\begin{lemma}\label{lm:3.2}
Let $\{w_{n}\}\subset V$ be a sequence such that
$$
\| w_{n}\|^2_{H^1_{0,L}(\mathcal{C}_{\Omega})}
=\int_{\mathcal{C}_{\Omega}}k_{\alpha}y^{1-\alpha}|\nabla w_{n}|^2\,dx\,dy
\to S_{\alpha,N}
$$
as $n\to \infty $,  then $ \operatorname{dist}(\beta(w_{n}),\Omega )\to 0$, as 
$n \to\infty$.
 \end{lemma}

\begin{proof} 
Suppose by contradiction that $ \operatorname{dist}(\beta(w_{n}),\Omega )\not\to 0$ 
as $n \to\infty $.
We may verify that $\{w_n\}$ is tight. By Lemma \ref{lm:3.1}, there exist 
sequences $\{\mu_k\}$ and $\{\nu_k\}$ such that
  \begin{gather}\label{e3.2}
S_{\alpha,N} =\lim_{n\to\infty}\| w_{n}\|^2
=k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}|\nabla w|^2\, dx\,dy
+\Sigma_{k\in I} \mu_{k}, \\
\label{e3.3}
 1=\lim_{n\to\infty}\int_{\Omega}\| u_{n}\|^2
=\int_{\Omega}|u|^{2^{\ast}_{\alpha}}\,dx+\Sigma_{k\in I} \nu_{k}.
\end{gather}
By the Sobolev inequality and Lemma \ref{lm:3.1},  from \eqref{e3.2} we deduce that
 $$
S_{\alpha,N} =\| w\|^2_{H^1_{0,L}(\mathcal{C}_{\Omega})}+\Sigma_{k\in I} \mu_{k}
\geq S_{\alpha,N}\|u\|_{L^{2^*_{\alpha}}(\Omega)}^2
+S_{\alpha,N} (\Sigma_{k\in I} \nu_{k})^{\frac{2}{2^*_{\alpha}}}.
$$
 Hence,
  \begin{equation}\label{e3.4}
  \|u\|_{L^{2^{\ast}_{\alpha}}(\Omega)}^2+ (\Sigma_{k\in I}
\nu_{k})^{\frac{2}{2^{\ast}_{\alpha}}}\leq 1 .
\end{equation}
Equations \eqref{e3.3} and \eqref{e3.4} imply either
 $\Sigma_{k\in I} \nu_{k}=0$ or $\|u\|_{L^{2^*_{\alpha}}(\Omega)}^{2^*_{\alpha}}=0$.

If $\Sigma_{k\in I} \nu_{k}=0$, that is 
$\|u\|^{2^*_{\alpha}}_{L^{2^*_{\alpha}}}(\Omega)=1$, the lower semi-continuity 
of norms yields
$$
S_{\alpha,N}\geq \| w\|^2_{H^1_{0,L}(\mathcal{C}_{\Omega})}
=\frac{k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}|\nabla w|^2\,dx\,dy}
{(\int_{\Omega}|u|^{2^*_{\alpha}}\,dx)^{\frac{2}{2^*_{\alpha}}}}.
$$
While by the Sobolev trace inequality,
$$
S_{\alpha,N}\leq \frac{k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}
|\nabla w|^2\,dx\,dy}
{(\int_{\Omega}|u|^{2^{\ast}_{\alpha}}\,dx)^{\frac{2}{2^*_{\alpha}}}},
$$
it then implies that $S_{\alpha,N}$ is achieved, which is a contradiction 
to the fact that $S_{\alpha,N}$ is not achieved unless 
$\mathcal{C}_{\Omega}=\mathbb{R_{+}}^{N+1}$. Thus,
 $\|u\|^{2^*_{\alpha}}_{L^{2^*_{\alpha}}}(\Omega)\neq 1$. Consequently,
 $\Sigma_{k\in I} \nu_{k}=1$ and $u = 0$.
Furthermore, by the uniqueness of the extension of $u$, we have $w=0$. 
Now, it is standard to show that  $\nu$ is concentrated at a single $x_0$
of  $\bar\Omega$. So we have
$$
\beta (w_{n})\to \int_{\Omega}x\, d\nu(x)=x_0\in\bar\Omega,
$$  
this is a contradiction.
\end{proof}

Since $\Omega$ is a smooth bounded domain of $\mathbb{R^{N}}$, we choose 
$r>0$ small enough so that
$$
\Omega_{r}^{+}=\{x\in \mathbb{R^{N}}: \operatorname{dist}(x ,\Omega))< r\}
\quad \text{and} \quad \Omega_{r}^{-}
=\{x\in \Omega: \operatorname{dist}(x ,\partial\Omega)> r\} 
$$
are homotopically equivalent to $\Omega$. Moreover we assume that the ball 
$B_r(0)\subset\Omega$, and then
 $ \mathcal{C}_{B_r(0)}:=B_r(0)\times(0,+\infty)\subset \mathcal{C}_{\Omega}$. 
We define
$$ 
V_0:=\{w \in H^1_{0,L}(\mathcal{C}_{B_r(0)}):
\int_{\mathcal{C}_{B_r(0)}}w_{+}^{2^*_{\alpha}}(x,0)\,dx=1 \}\subset V
$$
as well as
$$
Q_0=\inf_{w\in V_0} \varphi_{\lambda}(w).
$$
Denote by $\varphi_{\lambda}^{Q_0}:=\{w\in V: \varphi_{\lambda}(w)<Q_0\}$ 
the level set below $Q_0$. We may verify
as in Lemma \ref{lm:3.2} that $Q_0 <S_{\alpha,N}$.

\begin{lemma}\label{lm:3.3}
There exists a $\lambda^*$, $0<\lambda^*<\lambda_{1}$ such that for 
$0<\lambda<\lambda^*$, if $w\in\varphi_{\lambda}^{Q_0}$, then 
$ \beta(w)\in\Omega_{r}^{+}$.
 \end{lemma}

\begin{proof}
By H\"{o}lder's inequality, for every $w\in V$,
 $$
\int_{\Omega}|w(x,0)|^2\,dx
\leq \Big(\int_{\Omega}|w(x,0)|^{2^*_{\alpha}}\,dx\Big)^{\frac{2}{2^*_{\alpha}}}
 |\Omega|^{\alpha/N}
=|\Omega|^{\alpha/N}.
$$
Let $\lambda^* =\frac{\epsilon}{|\Omega|^{\alpha/N}}$. 
If $0<\lambda<\lambda^*$ and $w\in\varphi_{\lambda}^{Q_0}$, we have
$$ 
\|w\|^2\leq\lambda\int_{\Omega}|w(x,0)|^2\,dx
+Q_0\leq\lambda^*|\Omega|^{\alpha/N}+S_{\alpha,N}
= S_{\alpha,N}+\epsilon.
$$
Therefore, we conclude by Lemma \ref{lm:3.2} that $\beta(w)\in\Omega_{r}^{+}$.
\end{proof}

Now, we establish the relation of category between the domain $\Omega$ 
and the level set $\varphi_{\lambda}^{Q_0}$.

\begin{lemma}\label{lm:3.4}
If $N\geq 2\alpha$ and $0<\lambda<\lambda^*$,  then we have  
$\operatorname{cat}_{\varphi_{\lambda}^{Q_0}}\varphi_{\lambda}^{Q_0}
\geq \operatorname{cat}_{\Omega}(\Omega)$.
  \end{lemma}

\begin{proof} 
Let $w_0\in H^1_{0,L}(\mathcal{C}_{B(0,r)})$ be a minimizer of $Q_0$.
Hence, we may assume that $w_0>0$ is cylinder symmetric and
$\|w_0\|_{L^{2^*_{\alpha}}(B_r(0))}=1$,
$$
Q_0 =\int_{\mathcal{C}_{B_r(0)}}k_{\alpha}y^{1-\alpha}|\nabla w_0|^2\,dx\,dy
-\lambda\int_{B_r(0)}|w_0(x,0)|^2\,dx.
$$
For $z\in\Omega_{r}^{-}$, we define 
$\gamma: \Omega_{r}^{-}\to \varphi_{\lambda}^{Q_0}$ by
$$
\gamma(z) =
\begin{cases}
w_0(x-z,y),  &(x,y)\in B_r(z)\times (0,+\infty),\\
0, &(x,y)\notin B_r(z)\times (0,+\infty).
\end{cases}
$$
Since $w_0(x,0)$ is a radial function,
$$
\beta\circ\gamma(z) =\int_{B_r(z)}x(w_0)^{2^*_{\alpha}}_{+}(x-z,0))\,dx
=\int_{B_r(0)}x(w_0)^{2^*_{\alpha}}_{+}(x,0)\,dx+z
=z.
$$
Hence, $\beta\circ\gamma=id$.

Assume that $\varphi_{\lambda}^{Q_0}=A_{1}\cup A_{2}\cup \dots \cup A_{n}$, 
where $A_{j},  j=1,2\dots n$, is closed and contractible
in $\varphi_{\lambda}^{Q_0}$, i.e. there exists 
$h_{j}\in C([0,1]\times A_{j},\varphi_{\lambda}^{Q_0})$ such that, 
for every $u$, $ v\in A_{j}$,
$$ 
h_{j}(0,u)=u,\quad h_{j}(1,u)=h_{j}(1,v).
$$

Let $B_{j} :=\gamma^{-1}(A_{j})$, $1\leq j\leq n$. The sets $B_{j}$ are 
closed and $\Omega_{r}^{-}=B_{1}\cup B_{2}\dots\cup B_{n}$. 
By Lemma \ref{lm:3.3}, we know $\beta (h_{j}(t,\gamma(x)))\in\Omega_{r}^{+}$. 
Using the deformation $g_{j}(t,x)=\beta (h_{j}(t,\gamma(x)))$, we see that 
$B_j$ is contractible in $\Omega_r^+$. Indeed,
for every $x$, $y\in B_{j}$, there exist $\gamma(x)$, $\gamma(y)\in A_{j}$ 
such that
\begin{gather*}
g_{j}(0,x)=\beta(h_{j}(0,\gamma(x)))=\beta(\gamma(x))=x,\\
g_{j}(1,x)=\beta(h_{j}(1,\gamma(x)))=\beta(h_{j}(1,\gamma(y)))=g_{j}(1,y).
\end{gather*}
It follows that 
$\operatorname{cat}_{\varphi_{\lambda}^{Q_0}}\varphi_{\lambda}^{Q_0}
\geq \operatorname{cat}_{\Omega_{r}^{+}}(\Omega_{r}^{-}) 
=\operatorname{cat}_{\Omega}(\Omega)$.
\end{proof}



\begin{lemma}\label{lm:3.5}
If $\varphi_{\lambda}|_{V}$ is bounded from below and satisfies the $(PS)_{c}$ 
condition for any 
$$
c \in [\inf_{w\in V}\varphi_{\lambda}, Q_0],
$$
then $\varphi_{\lambda}|_{V}$ has a minimum and level set
 $\varphi_{\lambda}^{Q_0}$ contains  at least 
$\operatorname{cat}_{\varphi_{\lambda}^{Q_0}}\varphi_{\lambda}^{Q_0}$
 critical points of $\varphi_{\lambda}|_{V}$.
  \end{lemma}

The proof of the above lemma can be found in \cite{W}.



\begin{proof}[Proof of Theorem \ref{thm:1.1}] 
 By Lemma \ref{lm:3.5}, for $0<\lambda<\lambda^*$,  the level 
set $\varphi_{\lambda}^{Q_0}$ contains  at least
 $m:=\operatorname{cat}_{\varphi_{\lambda}^{Q_0}}\varphi_{\lambda}^{_0}$ 
critical points $w_{1}$, $w_{2},\dots ,w_{m}$ of $\varphi_{\lambda}|_{V}$.

For $j=1,2,\dots,m$, there exist $\mu_{j}\in\mathbb{R}$ such that, for 
$h\in H_{0,L}^1(\mathcal{C}_\Omega)$,
$$
k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha} 
\nabla w _{j}\nabla h\, dx\,dy-\lambda\int_{\Omega}wh \,dx
-\mu_{j}\int_{\Omega}(w_{j}^{+})^{2^{\ast}_{\alpha}-1} h\,dx=0.
$$
Choosing $h=w_{j}^{-}$, we have
$$ 0= k_{\alpha}\int_{C_{\Omega}}y^{1-\alpha} |\nabla w _{j}^{-}|^2dx\,dy
-\lambda\int_{\Omega}|w_{j}^{-}|^2dx.$$
Since $0<\lambda<\lambda_{1}$, it implies $w_{j}^{-}=0$ and
$$
k_{\alpha}\int_{\mathcal{C}_{\Omega}}y^{1-\alpha} |\nabla w _{j} |^2\,dx\,dy
-\lambda\int_{\Omega}|w_{j}|^2\,dx-\mu_{j}\int_{\Omega}(w_{j}^{+}
)^{2^*_{\alpha}}\, dx=0.
$$
Therefore, $\mu_{j}=\varphi_{\lambda}(w_{j})$ and 
$v_{j}:=\mu_{j}^{\frac{N-\alpha}{2\alpha}}w_{j}$ is a positive solution of
 \eqref{e1.4}, $tr_{\Omega}(v_{j})$ is a solution
 of \eqref{e1.1}. By Lemma \ref{lm:3.4},  problems \eqref{e1.4}
and \eqref{e1.1} have at least $\operatorname{cat}_{\Omega}(\Omega)$
positive solutions.
\end{proof}


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\end{document}