\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 234, pp. 1--26.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/234\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for nonlinear equations involving
 the square root of the Laplacian}

\author[Y. Chen, J. Su, H. Yan \hfil EJDE-2017/234\hfilneg]
{Yutong Chen, Jiabao Su, Huanhong Yan}

\address{Yutong Chen \newline
School of Mathematical Sciences,
Capital Normal University, Beijing 100048, China}
\email{chenyutong@cnu.edu.cn}

\address{Jiabao Su \newline
 School of Mathematical Sciences,
Capital Normal University,
Beijing 100048, China}
\email{sujb@cnu.edu.cn}

\address{Huanhong Yan \newline
 School of Mathematical Sciences,
 Capital Normal University,
Beijing 100048,  China}
\email{1464938716@qq.com}

\dedicatory{Communicated by Paul Rabinowitz}

\thanks{Submitted June 21, 2017. Published September 29, 2017.}
\subjclass[2010]{35J10, 35J65 58E05}
\keywords{Fractional Laplacian; variational methods; critical group;
\hfill\break\indent  Morse theory}

\begin{abstract}
 This paper deals with the existence and multiplicity results for fractional
 problem involving the square root of the Laplacian $A_{1/2}$ in a bounded
 domain with zero Dirichlet boundary conditions by Morse theory and critical
 groups for a $C^{1}$ functional at both isolated critical points and infinity.
\end{abstract}

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

\section{Introduction}

 This article concerns the existence and multiplicity of nontrivial weak
 solutions to nonlinear equations involving a non-local positive operator,
 the square root of the Laplacian in a bounded domain with zero Dirichlet
 boundary condition. Precisely, we study the fractional problem
\begin{equation} \label{eP}
 \begin{gathered}
A_{1/2} u=f(u) \quad x\in\Omega,\\
 u=0 \quad x \in \partial\Omega,
\end{gathered}
\end{equation}
 where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $N\geqslant 2$,
and the nonlinearity $f:\mathbb{R}\to\mathbb{R}$ is a continuous function
that satisfies the condition
 \begin{enumerate}
 \item[(A1)] $f(0)\equiv 0$ and there exist $a>0$
and $1\leqslant p < 2^\sharp:=\frac{2N}{N-1}$ such that
 $$
|f(t)| \leqslant a (1 + |t|^{p-1}) \quad \text{for all } t\in \mathbb{R}.
$$
 \end{enumerate}
 According to \cite{CT2010-AD}, the operator $A_{1/2}$ is regarded as the square
 root of the Laplacian operator $-\Delta$ and is defined as follows.
Let $\{\lambda_{j}, \varphi_{j}\}_{j=1}^\infty$ be the eigenvalues and the
corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\Omega$
with zero Dirichlet boundary data on $\partial\Omega$;
 that is, $\int_\Omega \varphi_j \varphi_k dx = \delta_{j,k}$ and
 \begin{equation} \label{e1.1}
\begin{gathered}
-\Delta \varphi_{j}=\lambda_{j} \varphi_{j} \quad x\in\Omega,\\
 \varphi_{j}=0 \quad x \in \partial\Omega.
\end{gathered}
\end{equation}
 For $u \in H_{0}^{1}(\Omega)$ with
 $$
u(x)=\sum_{j=1}^{\infty}\alpha_{j}\varphi_{j}(x), \quad x \in \Omega,
$$
the operator $A_{1/2}$ is defined by
$$
A_{1/2}u:=\sum_{j=1}^{\infty}\alpha_{j}\lambda_{j}^{1/2}\varphi_{j}.
$$
It is proved in \cite{CT2010-AD} that the operator
 $A_{1/2}$ is self-adjoint and positive definite and
$\{\lambda_{j}^{1/2}, \varphi_j\}_{j=1}^\infty$ are the eigenvalues
and the corresponding eigenfunctions of $A_{1/2}$ on $\Omega$. Precisely,
 one has that
 \begin{equation} \label{e1.2}
\begin{gathered}
 A_{1/2} \varphi_{j}= \mu_j \varphi_{j} \quad x\in\Omega,\\
 \varphi_{j}=0 \quad x \in \partial\Omega,
\end{gathered}
\end{equation}
 here and in the sequel we denote by
 \begin{equation} \label{e1.3}
\mu_{j}:=\lambda_{j}^{1/2}, \quad j\in \mathbb{N}
 \end{equation}
the $j$-th eigenvalue of the operator $A_{1/2}$.
The precise mathematical description and basic properties of the
operator $A_{1/2}$ will be recalled in the next section.

 It should be pointed out that the operator $A_{1/2}$ is different from
the integro-differential operator $(-\Delta)^s$ with $s=1/2$, where
 $(-\Delta)^s$($0<s<1$) is defined, up to a constant, as
 $$
-(-\Delta)^su(x):= \int_{\mathbb{R}^N}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}dy,
\quad x\in\mathbb{R}^N
$$
and is the infinitesimal generators of L\'evy stable diffusion processes
 (see \cite{Bertoin1996}). In \cite{sv2014-144} the authors showed that
the operator $A_{1/2}$ depends on the domain $\Omega$ considered,
since its eigenfunctions and eigenvalues depend on $\Omega$, while the
integral one $(-\Delta)^{1/2}$ evaluated at some point is independent of
 the domain in which the equation is set. Besides, the eigenvalues and
eigenfunctions of these two fractional operators behave quite different.

The fractions of the Laplacian, such as the square root of the Laplacian
$A_{1/2}$ considered in the present paper, appear in flames propagation and
chemical reactions in liquids, population dynamics, geophysical fluid dynamics,
anomalous diffusions in plasmas, and American options in finances
 (see \cite{app2004,Garroni2005,V2009}).

 Nonlinear equations involving the fractional Laplacian have attracted much
attention in the recent years. A lot of interest has been devoted to the
fractional Laplacian problems with various nonlinearities in getting the
existence, non-existence and regularity results as well as the qualitative
properties, see
\cite{Ambrosio2016,Arcoya2012,Barrios2012-JDE, Bradle2013, CS2014,
CS2015-TAMS,CT2010-AD, Caffarelli2010,Caffarelli2010-2,Capella2011,
CW2013, CW2014, silvestre2007,tan2011,tan2013,yu2012}
and the references therein.

 Through the Dirichlet-to-Neumann map due to Stein(\cite{stein}) on
$\Omega$, Cabr\'e and Tan in their well-known work \cite{CT2010-AD}
constructed a framework by transforming the nonlocal problem \eqref{eP}
to a local problem equivalently in the cylinder
$\mathcal{C}= \Omega\times(0, \infty)$ with mixed boundary data which
has variational structure so that the classical variational methods work well.
Under such a framework from \cite{CT2010-AD}, the existence of a positive
solution of \eqref{eP} for $f(u)=|t|^{q-1}t$ with $1<q<\frac{N+1}{N-1}$
was obtained in \cite{CT2010-AD} by constrained minimization method,
Tan studied in \cite{tan2011} the existence of a positive solution of
\eqref{eP} with critical nonlinearity case of
$f(t)=\mu t+ |t|^{\frac{2}{N-1}}t$ by the mountain pass theorem, and
in \cite{yu2012} Nehari manifold method was applied to get the the
existence of solutions and multiple solutions of \eqref{eP} for
$f(t)= \mu t + b(x) |t|^{q-1} t $ with $0<q<\frac{N+1}{N-1}$ and
sign-changing weight $b(x)$.

 The aim of the present paper is to establish the existence and multiplicity
results for \eqref{eP} for general nonlinear function $f$ with subcritical growth.
 The problem \eqref{eP} has admitted a trivial solution $u=0$ because
$f(0) \equiv 0$, we are interested in finding nontrivial weak solutions
of \eqref{eP}. The existence of nontrivial weak solutions of \eqref{eP}
depends mainly upon the behaviors of the nonlinear term
 $f$ or its primitive $F(t)=\int_{0}^{t}f(s)ds$ near infinity and near zero.

 Now we state the assumptions on the nonlinearity $f$ in our context.
 Near infinity we make the following assumptions.
\begin{enumerate}
 \item[(A2)] There exist $R>0$ and $\theta>2$ such that
\begin{equation} \label{e1.4}
0<\theta F(t)\leqslant f(t)t \quad \text{for } |t|\geqslant R.
 \end{equation}

\item[(A3)] For some eigenvalue $\mu_{m}$ of $A_{1/2}$ with $m>1$ there
exist the limits
\begin{gather}
\lim _{|t|\to \infty} \frac{f(t)}{t}=\mu_{m},\label{e1.5} \\
\lim _{|t|\to \infty}\pm(f(t)t-2F(t))= +\infty.\label{e1.6}%_{\pm}
\end{gather}

 \item[(A4)] There exist $\mu < \mu_1$ and $C>0$ such that
\begin{equation}
F(t)\leqslant\frac{1}{2}\mu t^{2}+C \quad \text{for all }
 t \in \mathbb{R}.\label{e1.7}
\end{equation}

 \item[(A5)] There exist the limits
\begin{gather}
\lim _{|t|\to \infty} \frac{2F(t)}{t^{2}}=\mu_{1},\label{e1.8} \\
\lim _{|t|\to \infty}(2F(t)-\mu_1 t^2)=-\infty.\label{e1.9}
\end{gather}
 \end{enumerate}
 Near the origin we make the following assumptions.
\begin{enumerate}

 \item[(A6)] There exist $\delta>0$ and $\tau\in(1, 2)$ such that
\begin{gather}
f(t)t>0 \quad \text{for } 0<|t|\leqslant \delta,\label{e1.10}\\
\tau F(t)-f(t)t\geqslant0 \quad \text{for } |t|\leqslant \delta.\label{e1.11}
\end{gather}

 \item[(A7)] There exist $\delta>0$ and $k\geqslant1$ such that for
two different adjacent eigenvalues $\mu_{k}<\mu_{k+1}$ of $A_{1/2}$, it holds that
\begin{equation}
\mu_{k}t^{2}\leqslant 2F(t)\leqslant\mu_{k+1}t^{2} \quad\text{for }
 |t|\leqslant \delta,\label{e1.12}
\end{equation}

 \item[(A8)] There exist $\delta>0$ such that
\begin{equation}
 2F(t)\leqslant\mu_{1} t^{2} \quad \text{for }
 |t|\leqslant \delta. \label{e1.13}
\end{equation}
 \end{enumerate}

 The main results of this article are the following theorems for equations
 driven by the square root of the Laplacian. The first theorem is
related to the existence of one nontrivial weak solution of \eqref{eP}.

 \begin{theorem}\label{thm1.1}
 Assume {\rm (A1)}.
 Then  problem \eqref{eP} admits at least one nontrivial weak solution
in each of the following cases:
\begin{itemize}
\item[(a)] {\rm (A2)} and {\rm (A7)},
\item[(b)] {\rm (A2)} and {\rm (A8)},
\item[(c)] {\rm (A3)} and {\rm (A6)},
\item[(d)] {\rm (A3)} and {\rm (A8)},
\item[(e)] {\rm (A4)} and {\rm (A6)},
\item[(f)] {\rm (A5)} and {\rm (A6)}.
\end{itemize}
 \end{theorem}

 In the second theorem we establish the multiplicity of nontrivial solutions
of \eqref{eP}.

 \begin{theorem} \label{thm1.2}
Assume {\rm (A1)}. Then  problem \eqref{eP} admits at least two nontrivial
 weak solutions in each of the following cases:
\begin{itemize}
\item[(a)] {\rm (A4)} and {\rm (A7)},
\item[(b)] {\rm (A5)} and {\rm (A7)}.
\end{itemize}
 \end{theorem}

Now we give some remarks about the conditions presented above.

 We first look at the conditions near infinity.
 The condition (A2) is the well-known Ambrosetti-Rabinowitz
superquadratic condition at infinity introduced in the pioneering paper
\cite{ar1973} and has been used extensively in the literature in dealing
with superlinear variational problems. We mention a famous work \cite{Wang1991}
 by Wang where the condition (A2) was exploited to describe the topological
property of the energy functional at infinity and a third nontrivial solution
for superlinear elliptic equations was obtained via Morse theory.
In the condition (A3), \eqref{e1.5} means that  problem \eqref{eP} is
completely resonant at the eigenvalue $\mu_m$ of $A_{1/2}$ near infinity,
 while \eqref{e1.6} is so-called the non-quadratic conditions
 (see \cite{costa1994}). We regard the condition (A4) as a weak version of
sub-quadratic condition since it includes $\lim_{|t|\to \infty} 2F(t)/{t^2}=0$
or $\lim_{|t| \to \infty} f(t)/t= 0$ as the special case.
 The condition (A5) means  problem \eqref{eP} is resonant near infinity at
$\mu_1$ from the left side.
 In this case we use \eqref{e1.9} that is weaker than the case + in \eqref{e1.6}.

 Next we look at the conditions near zero. The conditions (A6)
 means that the function $f$ is superlinear near zero as which implies
$\lim_{t\to 0} 2F(t)/t^2 = \infty$. This condition was introduced in
\cite{liuwu1997} and a similar case was seen in \cite{moroz1997}.
The condition (A7) means that  problem \eqref{eP} is resonant near
zero between two consecutive eigenvalues of $A_{1/2}$. This condition
 was first introduced in \cite{lps2001}. The condition (A8)
 means that  problem \eqref{eP} is resonant near zero at $\mu_1$
from the left side.

 Theorems \ref{thm1.1} and \ref{thm1.2} will be proved by applying
 the infinite dimensional 
Morse theory to the fractional framework.
 Because of the nonlocal feature of  problem \eqref{eP} on $\Omega$, 
it is difficult for us to apply Morse theory directly.
 Instead, we apply Morse theory to an extended local problem in the 
cylinder $\mathcal{C}$ which is equivalent to \eqref{eP} according to 
the framework built in \cite{CT2010-AD}. By studying the variational 
functional corresponding to the extended local problem in the 
cylinder $\mathcal{C}$ with Morse theory and critical groups at zero and 
at infinity, we prove Theorem \ref{thm1.1} for the existence of one nontrivial 
weak solutions of \eqref{eP}. 
The multiplicity result in Theorem \ref{thm1.2} will be obtained by applying a 
three critical point theorem in \cite{liu-su2001}.

 The article is organized as follows. 
In Section 2, we present the functional space related to  problem \eqref{eP}
 together with the basic properties about the operator $A_{1/2}$. 
Then we recall some abstract results about Morse theory and critical groups.
 In Section 3, we satisfy the compactness of the functional and give the
computations of critical groups at infinity. 
In Section 4, we compute the critical groups at zero. 
In Section 5, we give the proofs of Theorems \ref{thm1.1} and \ref{thm1.2}.

 \section{Preliminaries}

 In this section we will give the preliminaries for the variational settings 
related to  problem \eqref{eP} and some abstract results in Morse theory.

 \subsection{Functional spaces and the operator $A_{1/2}$}
 We first recall briefly the functional framework built in \cite{CT2010-AD}.
 Denote the upper half space in $\mathbb{R}^{N+1}$ by
 $$ 
\mathbb{R}_+^{N+1} =\{(x, y): \ x\in \mathbb{R}^N, \ y>0 \}, 
$$ 
and the half cylinder standing on $\Omega$ by 
$\mathcal{C}=\Omega\times (0,+\infty)$ and its lateral boundary by
$\partial_{L}\mathcal{C}=\partial \Omega\times (0,\infty)$.
Consider the Sobolev space of functions with trace vanishing on 
$\partial_{L}\mathcal{C}$:
$$
H_{0,L}^{1}(\mathcal{C})=\Big\{v\in L^2(\mathcal{C}) :
 v=0  \text{ on }  \partial_{L}\mathcal{C}, \;
 \int_{\mathcal{C}}|\nabla v|^2\,dx\,dy<\infty\Big\}
.$$
Then $H_{0,L}^{1}(\mathcal{C})$ is a Hilbert space with the scalar product
 $$
\langle v, w\rangle=\int_{\mathcal{C}} \nabla v \nabla w \,dx\,dy 
$$ 
and the norm
 $$
\|v\|=\Big(\int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy \Big)^{1/2}.
$$
 From \cite[Lemmas 2.4 and 2.5]{CT2010-AD} we get the following embedding results:

 \begin{proposition}\label{prop2.1} 
The embedding from $H_{0,L}^{1}(\mathcal{C}) $ into $L^q(\Omega)$ is 
continuous for all $q \in [1, \frac{2N}{N-1}]$ and is compact for all 
$q\in[1, \frac{2N}{N-1})$. Moreover, there is $c_q>0$ such that 
\begin{equation}
\Big(\int_{\Omega\times\{0\}}|v(x, 0)|^q dx\Big)^{1/q}
\leqslant c_q \Big(\int_{\mathcal{C}}|\nabla v|^2 \,dx\,dy\Big)^{1/2} \quad
\text{for all }  v\in H_{0,L}^{1}(\mathcal{C}).\label{e2.1}
\end{equation}
 \end{proposition}

 Denote by $\operatorname{tr}_{\Omega}$ the trace operator on
$\Omega \times \{0\}$ for functions in $H_{0,L}^{1}(\mathcal{C})$:
 $$
\operatorname{tr}_{\Omega}v: = v(\cdot, 0), \text{ for }v \in H_{0,L}^{1}(\mathcal{C}).
$$
 Let $\mathcal{V}_{0}(\Omega)$ be the space of all traces on 
$\Omega\times \{0\}$ of functions in $H_{0,L}^{1}(\mathcal{C})$;
 that is, 
$$
\mathcal{V}_{0}(\Omega):= \big\{u=\operatorname{tr}_{\Omega}v :
 v\in H_{0,L}^{1}(\mathcal{C})\big\}. 
$$
 Then by \cite[Lemma 2.10]{CT2010-AD}, $\mathcal{V}_{0}(\Omega)$ can be 
characterized as
\begin{equation}
\mathcal{V}_{0}(\Omega)= \Big\{u\in L^{2}(\Omega) :
u=\sum_{j=1}^{\infty}\alpha_{j}\varphi_{j}  \text{ satisfies }
 \sum_{j=1}^{\infty}\alpha_{j}^{2}\lambda_{j}^{1/2}<+\infty \Big\}\label{e2.2}
\end{equation}
and the space $H_{0,L}^{1}(\mathcal{C})$ can be characterized as
 (see the proof of\cite[ Lemma 2.10]{CT2010-AD})
\[
H_{0,L}^{1}(\mathcal{C}) = \Big\{v \in L^{2}(\mathcal{C}) :
 v(x, y)=\sum_{j=1}^{\infty}\alpha_{j}\varphi_{j}\exp(-\lambda_{j}^{1/2}y)
 \text{ with }  \sum_{j=1}^{\infty}\alpha_{j}^{2}\lambda_{j}^{1/2}<+\infty \Big\}.
%\label{e2.3}
\]
 Where the pair $\{\lambda_j, \varphi_j\}_{j\in \mathbb{N}}$ are the eigenvalue
and the corresponding eigenfunction of $-\Delta$ on $\Omega$ with zero boundary
value on $\partial \Omega$, as stated in \eqref{e1.1}.

 For a given function $u\in \mathcal{V}_0(\Omega)$, its harmonic extension $v$
 to the cylinder $\mathcal{C}$ is the weak solution of  the problem
\begin{equation}
\begin{gathered}
-\Delta v=0 \quad \text{in }  \mathcal{C},\\
 v=0 \quad \text{on }  \partial_L\mathcal{C},\\
 v=u \quad \text{on }  \Omega \times \{0\}.
\end{gathered}\label{e2.4}
\end{equation}
The idea of the harmonic extension was introduced in the pioneering
 work of Caffarelli-Silvestre\cite{Caffarelli2007} where the fractional
 Laplacian in the whole space was dealt with.

 The definition and properties of the operator $A_{1/2}$ are stated as follows.

 \begin{proposition}[\cite{CT2010-AD}]\label{prop2.2} 
For $u= \sum_{j=1}^{\infty}\alpha_{j}\varphi_{j}\in \mathcal{V}_{0}(\Omega)$, 
there exists a unique harmonic extension $v$ in $\mathcal{C}$ of $u$ such that 
$v\in H_{0,L}^{1}(\mathcal{C})$, and it is given by the expansion
\begin{equation}
v(x,y)=\sum_{j=1}^{\infty}\alpha_{j}\varphi_{j}(x)\exp(-\lambda_{j}^{1/2}y),\quad
\text{for all }  (x,y)\in \mathcal{C}.\label{e2.5}
\end{equation}
 The operator $A_{1/2}: \mathcal{V}_{0}(\Omega)\to \mathcal{V}_{0}^{*}(\Omega)$
is given by the Dirichlet-to-Neumann map
\begin{equation}
A_{1/2}u:=\frac{\partial v}{\partial \nu}\Big|_{\Omega \times \{0\}},\label{e2.6}
\end{equation}
where $\mathcal{V}_{0}^{*}(\Omega)$
 is the dual space of $\mathcal{V}_{0}(\Omega)$ and where $\nu$ is
the unit outer normal to $\mathcal{C}$ at $\Omega\times \{0\}$.
 We have
\begin{equation}
A_{1/2}u=\sum_{j=1}^{\infty}\alpha_{j}\lambda_{j}^{1/2}\varphi_{j},\label{e2.7}
\end{equation}
and that $A_{1/2}\circ A_{1/2}$ is equal to $-\Delta $ in $\Omega$
with zero Dirichlet boundary values on $\partial \Omega$.
 The inverse $A_{1/2}^{-1}$ is the unique positive square root of the
inverse Laplacian $(-\Delta )^{-1}$ in $\Omega$ with zero Dirichlet boundary
values on $ \partial \Omega$.
 \end{proposition}

 Now we consider the linear eigenvalue problem
\begin{equation}
\begin{gathered}
A_{1/2} u = \mu u \quad \text{in }  \Omega,\\
 u=0 \quad \text{on }  \partial\Omega.
\end{gathered}\label{e2.8}
\end{equation}
 By the definition of $A_{1/2}$, we see that a nontrivial function
 $u \in \mathcal{V}_0(\Omega)$
 is an eigenfunction associated to the eigenvalue $\mu$ if and only
if the harmonic extension $v$ of $u$ to the cylinder $\mathcal{C}$ satisfies
\begin{equation}
\begin{gathered}
-\Delta v=0 \quad  \text{in }  \mathcal{C},\\
 v=0 \quad  \text{on }\partial_L\mathcal{C},\\
 \frac{\partial v}{ \partial \nu} = \mu u \quad \text{on } \Omega \times \{0\}.
\end{gathered} \label{e2.9}
\end{equation}
 We have that $\{\lambda_j^{1/2}, \varphi_j\}_{j\in \mathbb{N}}$
are the eigenvalues and the corresponding eigenfunctions of \eqref{e2.8}
 (see \cite[Lemma 2.13]{CT2010-AD}). Setting
\begin{equation}
\mu_{j}=\lambda_{j}^{1/2} \quad \text{and} \quad
e_{j}(x,y)=\varphi_{j}(x)\exp(-\mu_j y) \quad \text{for all }  j\in \mathbb{N}.
\label{e2.10}
\end{equation}
Then all the pairs $\{\mu_j, e_j\}_{j\in \mathbb{N}}$ satisfy \eqref{e2.9}:
for all $j\in \mathbb{N}$,
\begin{equation}
\begin{gathered}
-\Delta e_j=0 \quad \text{in } \mathcal{C},\\
 e_j=0 \quad  \text{on } \partial_L\mathcal{C},\\
 \frac{\partial e_j}{ \partial \nu} = \mu_j e(\cdot, 0)= \mu_j \varphi_j \quad
 \text{on }  \Omega \times \{0\},
\end{gathered}\label{e2.11}
\end{equation}
 The eigenfunction sequence $\{e_j\}_{j\in \mathbb{N}}$ forms an orthogonal
basis of $H_{0,L}^{1}(\mathcal{C})$. The eigenvalue sequence
$\{\mu_j\}_{j\in \mathbb{N}}$ has the following variational characterizations:
\begin{equation}
\mu_1 = \min_{v\in H_{0,L}^{1}(\mathcal{C})\setminus \{0\}}
\frac{\int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy}{\int_{\Omega} |v(x,0)|^{2}dx}
= \int_\mathcal{C} |\nabla e_1|^2 \,dx\,dy,\label{e2.12}
\end{equation}
and
\[
\mu_j = \min_{v \in \mathbb{P}_j \setminus \{0\}}
\frac{\int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy}{\int_{\Omega} |v(x,0)|^{2}dx}
 = \int_\mathcal{C} |\nabla e_j|^2 \,dx\,dy,
\]
where
 $$
\mathbb{P}_j=\big\{v\in H^1_{0, L}(\mathcal{C}) :
 \langle v, e_i \rangle=0  \text{ for }  i=1, 2, \dots, j-1 \big\}.
$$
 Moreover, $\mu_{1}$ is simple and
 $0<\mu_{1}<\mu_{2}\leqslant \dots \leqslant \mu_{j} \leqslant \dots
\to \infty$ as $j\to \infty$,
 and that each $\mu_{j}$ has finite multiplicity.
 For $j\in \mathbb{N}$, let $\ell_j$ be the multiplicity of $\mu_j$; that is,
 $$
\mu_{j-1}<\mu_j=\mu_{j+1}=\dots=\mu_{j+\ell_j-1}<\mu_{j+\ell_j}.
$$
Set
\begin{gather*}
H^-(\mu_j) = \operatorname{span}\{e_1, \dots, e_{j-1}\}, \quad
H(\mu_j) = \operatorname{span}\{e_j, \dots, e_{j+\ell_j-1}\}, \\
H^+(\mu_j) =\overline{\operatorname{span}\{e_{j+\ell_j}, e_{j+\ell_j+1}, \dots,\}}
= \big[H^-(\mu_j)\oplus H(\mu_j)\big]^\perp.
\end{gather*}
 Then
\begin{equation}
 H^1_{0, L}(\mathcal{C})=H^-(\mu_j)\oplus H(\mu_j) \oplus H_j^+(\mu_j).\label{e2.13}
\end{equation}

 \begin{proposition}\label{prop2.3} 
 The following variational inequalities hold:
\begin{gather*}
\int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy 
 \leqslant \mu_{j-1} \int_{\Omega} |v(x,0)|^{2}dx \quad \text{for all } 
 v \in H^-(\mu_j),\label{e2.14} \\
\int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy 
 = \mu_{j} \int_{\Omega} |v(x,0)|^{2}dx, \quad \text{for all } 
 v \in H(\mu_j),\label{e2.15} \\
 \int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy \geqslant \mu_{j+\ell_j} 
\int_{\Omega} |v(x,0)|^{2}dx. \quad \text{for all }
 v\in H^+(\mu_j).\label{e2.16}
\end{gather*}
 \end{proposition}

 \subsection{Extended problem, weak solutions and variational formula}

 With the preliminaries in the previous subsection at hand, we turn to 
 problem \eqref{eP}.
 We say that a function $u \in \mathcal{V}_0(\Omega)$ is a {\it weak} 
solution of \eqref{eP} if the function $v \in H_{0,L}^{1}(\mathcal{C})$ 
with $\operatorname{tr}_\Omega v = v(\cdot,0)=u$ {\it weakly} 
solves the extended problem
\begin{equation} \label{ePt}
\begin{gathered}
-\Delta v=0 \quad \text{in } \mathcal{C},\\
 v=0 \quad  \text{on } \partial _{L}\mathcal{C},\\
 \frac{\partial v}{\partial \nu}=f(v(\cdot,0)) \quad \text{on }
 \Omega \times \{0\},
\end{gathered}
\end{equation}
that is the function $v$ satisfies the variational formula
\begin{equation}
 \int_{\mathcal{C}} \nabla v\nabla \phi \,dx\,dy
=\int_{\Omega}f(v(x,0))\phi(x,0) dx \quad \text{for all }
 \phi\in H_{0,L}^{1}(\mathcal{C}).\label{e2.17}
\end{equation}
 Observe that the extended problem \eqref{ePt}
 has a variational structure, indeed, it is the Euler-Lagrange equation
of the functional $\mathcal{J}: H_{0,L}^{1}(\mathcal{C}) \to \mathbb{R}$ defined by
\begin{equation}
\mathcal{J}(v)= \frac{1}{2}\int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy
- \int_{\Omega} F(v(x,0))dx, \quad v \in H_{0,L}^{1}(\mathcal{C}).\label{e2.18}
\end{equation}
 Since the nonlinear function $f$ satisfies the subcritical growth condition
(A1), by Proposition \ref{prop2.1},  the functional $\mathcal{J}$ is well-defined on
$H_{0,L}^{1}(\mathcal{C})$ and is of class $C^1$ with derivative given by
\begin{equation}
\langle \mathcal{J}'(v), \phi\rangle
=  \int_{\mathcal{C}} \nabla v\nabla \phi \,dx\,dy
-\int_{\Omega}f(v(x,0))\phi(x,0) dx.\label{e2.19}
\end{equation}
 Therefore critical points of $\mathcal{J}$ are exactly weak solutions of
 \eqref{ePt} and then the traces of critical points of $\mathcal{J}$
are exactly weak solutions to problem \eqref{eP}.

 We will apply Morse theory and critical groups to find critical 
points of $\mathcal{J}$.

\subsection{Preliminaries about Morse theory}

 In this subsection we collect some results on Morse theory for a $C^1$
 functional $\mathcal{J}$ defined on a Hilbert space $E$.

 Let $\mathcal{J}\in C^1(E, \mathbb{R})$. Denote for $c\in \mathbb{R}$
 $$
\mathcal{J}^c=\{z \in E : \mathcal{J}(z) \leqslant c\}, \quad
 \mathcal{K}_c = \{ z \in E : \mathcal{J}'(z)=0, \; \mathcal{J}(z)=c\}.
$$
 We say that the functional $\mathcal{J}$ possesses the deformation property 
at the level $c \in \mathbb{R}$  if for any $\bar \epsilon>0$ and any 
neighborhood $\mathcal{N}$ of $\mathcal{K}_c$,
 there are $\epsilon>0$ and a continuous deformation $\zeta: [0, 1]\times E \to E$ 
such that
\begin{itemize}
\item[(i)]  $\zeta(z, t) = z$ for either $t = 0$ or 
$z \not\in \mathcal{J}^{-1}([c-\bar{\epsilon}, c+\bar{\epsilon}])$;

\item[(ii)] $\mathcal{J}(\zeta(t, z))$ is nonincreasing in $t$ for any $z\in E$;

\item[(iii)] $\zeta(\mathcal{J}^{c+\epsilon}\setminus \mathcal{N})\subset 
\mathcal{J}^{c-\epsilon}$.
\end{itemize}
 We say that $\mathcal{J}$ possesses the deformation property if $\mathcal{J}$ 
possesses the deformation property at all $c\in \mathbb{R}$.

 We say that $\mathcal{J}$ satisfies the Palais-Smale condition at the level 
$c\in \mathbb{R}$ if any sequence
 $\{z_n\}\subset E $ satisfying $\mathcal{J}(z_n)\to c$ and 
$\mathcal{J}'(z_n) \to 0$ as
 $n\to \infty$ has a convergent subsequence. We say that $\mathcal{J}$ 
satisfies the the Palais-Smale condition condition
 if $\mathcal{J}$ satisfies the Palais-Smale condition at each $c\in \mathbb{R}$.

 We say that $\mathcal{J}$ satisfies the Cerami condition at the level 
$c\in \mathbb{R}$ if any sequence $\{z_n\}\subset E $ such that 
$\mathcal{J}(z_n)\to c$ and $(1+\|z_n\|)\|\mathcal{J}'(z_n)\| \to 0$ as
 $n\to \infty$ has a convergent subsequence. We say that $\mathcal{J}$ 
satisfies the Cerami condition if $\mathcal{J}$ satisfies the Cerami condition 
at any $c \in \mathbb{R}$.

 We note that if $\mathcal{J}$ satisfies the Palais-Smale condition or 
the Cerami condition then $\mathcal{J}$ possesses the deformation property
 (see \cite{chang1983,bl1997}).

 Let $\mathcal{K}=\{z \in E :\mathcal{J}'(z)=0 \}$. Assume that
 $\mathcal{J}(\mathcal{K})$ is bounded from below by $a \in \mathbb{R}$ and
 $\mathcal{J}$ possesses the deformation property at all $c \leqslant a$. 
The group $C_q(\mathcal{J}, \infty):=H_q(E, \mathcal{J}^a), q\in \mathbb{Z}$, 
is called the  $q$-th critical group of $\mathcal{J}$ at infinity (\cite{bl1997}), 
where $H_*(A, B)$ denotes a singular relative homology group
 of the pair $(A, B)$ with integer coefficients.

 Let $z_0$ be an isolated critical point of $\mathcal{J}$ with 
$\mathcal{J}(z_0)=c\in \mathbb{R}$, and $U$ be a neighborhood of $z_0$ such 
that $U\cap \mathcal{K}=\{z_0\}$.
 The group  $C_q(\mathcal{J}, z_0):=H_q(\mathcal{J}^c \cap U, 
\mathcal{J}^c \cap U \setminus \{z_0\}), q \in \mathbb{Z}$, is called 
the $q$-th critical group of $\mathcal{J}$ at $z_0$.

 Assume that $\mathcal{J}$ possesses the deformation property and $\mathcal{K}$ 
is a finite set. We have the following basic facts from Morse theory 
(see \cite{chang1993, mw1989, bl1997}). If $\mathcal{K}=\emptyset$ 
then $C_q(\mathcal{J},\infty)\cong0$ for all $q\in \mathbb{Z}$. 
Thus if $C_q(\mathcal{J}, \infty)\not\cong 0$ for some $q \in \mathbb{Z}$ 
then $\mathcal{K} \ne \emptyset$. Assume that $0\in \mathcal{K}$. 
If $\mathcal{K}=\{0\}$ then $C_q(\mathcal{J}, \infty) \cong C_q(\mathcal{J}, 0)$ 
for all $q\in \mathbb{Z}$.
 Thus if $C_q(\mathcal{J}, \infty) \not \cong C_q(\mathcal{J}, 0)$ for some 
$q\in \mathbb{Z}$ then $\mathcal{J}$ must have a critical point differing from $0$.
 Therefore the basic idea in applying Morse theory to find nonzero critical
 points of $\mathcal{J}$ is to compute critical groups both at infinity and at $0$.


The critical group $C_q(\mathcal{J}, \infty)$ can be computed partially when
 $\mathcal{J}$ has a saddle point geometry at infinity.

 \begin{proposition}[\cite{bl1997}] \label{prop2.4}
 Let $E$ be a Hilbert space such that $E= V_\infty \oplus W_\infty $ 
with $\ell =\dim V_\infty< \infty$. 
Let $\mathcal{J}\in C^1(E, \mathbb{R})$ possess the deformation property.
 Suppose that $\mathcal{J}$ satisfies
 \begin{itemize}
 \item[(i)] $\inf _{z\in W_\infty} \mathcal{J}(u)> -\infty$;
 \item[(ii)] $\mathcal{J}(z) \to -\infty$ as $\|z\|\to \infty$, $z\in V_\infty$.
 \end{itemize}
 Then $C_\ell(\mathcal{J},\infty)\not\cong 0$.
 \end{proposition}

 The critical group $C_q(\mathcal{J}, 0)$ can be computed partially when 
$\mathcal{J}$ has a local linking structure at zero.

 \begin{proposition}[\cite{liu1989}] \label{prop2.5}
Let $E$ be a Hilbert space such that $E= V_0\oplus W_0 $ with 
$\ell_0 =\dim V_0< \infty$. Let $\mathcal{J}\in C^1(E, \mathbb{R})$ 
possess the deformation property. Assume that $\mathcal{J}$ has an 
isolated critical point $z=0$ with $\mathcal{J}(0)=0$. 
If $\mathcal{J}$ has a local linking at $0$ with respect to
 $E= V_0\oplus W_0$ where $\ell_0 = \dim V_0<\infty$ i.e., 
there exists $\rho>0$ small such that
\begin{equation}
\mathcal{J}(z)\leqslant 0, \quad z\in V_0,\; \|z\|\leqslant \rho, \quad
\mathcal{J}(z)> 0, \; z\in W_0, \; 0<\|z\|\leqslant \rho.\label{e2.20}
\end{equation}
 Then $C_{\ell_0}(\mathcal{J},0)\not\cong 0$.
\end{proposition}

 In this subsection we recall a very general version of the famous three 
critical point theorem.

 \begin{proposition}[\cite{liu-su2001}] \label{prop2.6} 
 Let $E$ be a Hilbert space and let $\mathcal{J}\in C^1(E, \mathbb{R})$ 
possess the deformation property and be bounded from below. 
Assume that $\mathcal{J}$ has an isolated critical point $z_{*}\in E$ such that
 \begin{itemize}
 \item[(i)] $z_{*}$ is homological nontrivial, i.e., 
$C_q(\mathcal{J}, z_{*})\not\cong 0$ for some $q\in \mathbb{Z}$,
 \item[(ii)] $z_{*}$ is not the global minimizer of $\mathcal{J}$.
 \end{itemize}
 Then $\mathcal{J}$ has at least three critical points.
 \end{proposition}

We point out that the all above results on Morse theory are valid for 
$E$ being a Banach space.

 \section{Compactness and critical groups at infinity}

 We will prove Theorems \ref{thm1.1} and \ref{thm1.2}
 by studying the functional 
$\mathcal{J}$ defined by \eqref{e2.18}:
 $$
J(v)=\frac12\int_\mathcal{C} |\nabla v|^2\,dx\,dy
 -\int_{\Omega} F(v(x, 0))dx, \quad  v\in H_{0,L}^1(\mathcal{C}).
$$
 We will use Morse theory and critical groups computations to get the 
existence of nontrivial critical points of $\mathcal{J}$.
 First of all we study the bounded compactness of $\mathcal{J}$.
 We have the following result.

 \begin{lemma}\label{lem3.1} 
Let $f$ satisfy {\rm (A1)}. Then any bounded sequence 
$\{v_{n}\}\subset H_{0,L}^{1}(\mathcal{C})$ such that
\begin{equation}
\mathcal{J}'(v_n) \to 0\quad \text{in }  ( H_{0,L}^{1}(\mathcal{C}))^{*}
\text{ as } n\to\infty\label{e3.1}
\end{equation}
 has a convergent subsequence.
\end{lemma}

\begin{proof}
 Let $\{v_{n}\}\subset H_{0,L}^{1}(\mathcal{C})$ be bounded and satisfy 
\eqref{e3.1}.
 Since $ H_{0,L}^{1}(\mathcal{C})$ is a Hilbert space and then is 
reflexive, there is a subsequence of $\{v_{n}\}$, it is still denoted
by $\{v_{n}\}$, and there exists $v^* \in H_{0,L}^{1}(\mathcal{C})$, 
such that
 \begin{equation}
v_{n} \rightharpoonup v^*\text{ weakly  in $ H_{0,L}^{1}(\mathcal{C})$
 as }  n\to\infty.\label{e3.2}
\end{equation}
 By Proposition \ref{prop2.1}, up to a subsequence, it holds
\begin{equation}
\begin{gathered}
 \operatorname{tr}_{\Omega}v_n\to \operatorname{tr}_{\Omega}v^* \quad
 \text{strongly in } L^q (\Omega) \quad \forall  q \in [1, 2^{\sharp}),\\
 v_n(x,0)\to v^*(x,0) \quad \text{a.e.  in }  \Omega
 \end{gathered}\label{e3.3}
\end{equation}
 as $n\to \infty$, and there exists $\kappa_{q}\in L^q (\Omega)$
such that
\begin{equation}
|v_n(x,0)|\leqslant \kappa_{q}(x)\quad
\text{a.e. in $\Omega$ for any } n \in \mathbb{N}.\label{e3.4}
\end{equation}
By (A1), \eqref{e3.3}, \eqref{e3.4} and the Dominated Convergence Theorem,
we have
\begin{gather}
\lim_{n\to\infty} \int_{\Omega}f(v_n(x,0)) v_n(x,0)dx
=\int_{\Omega}f(v^*(x,0)) v^*(x,0)dx,\label{e3.5}\\
\lim_{n\to\infty} \int_{\Omega}f(v_n(x,0))v^*(x,0)dx=
\int_{\Omega}f( v^*(x,0)) v^*(x,0)dx.\label{e3.6}
\end{gather}
Since $\{v_n\}$ is bounded, by \eqref{e3.1} we have
\begin{equation}
 \langle\mathcal{J}'(v_{n}),v_{n}\rangle
 = \|v_{n}\|^{2}-\int_{\Omega}f(v_n(x,0)) v_n(x,0)dx\to 0\quad \text{as }
 n\to\infty.\label{e3.7}
\end{equation}
Consequently, from \eqref{e3.5}and \eqref{e3.7} we deduce that
\begin{equation}
\lim_{n\to\infty}\|v_{n}\|^{2} =\int_{\Omega}f( v^*(x,0))v^*(x,0)dx.\label{e3.8}
\end{equation}
Furthermore, using \eqref{e3.1} again, we have
\begin{equation}
  \langle \mathcal{J}'(v_{n}),v^*\rangle
= \langle v_{n},v^*\rangle -\int_{\Omega}f(v_n(x,0))v^*(x,0)dx \to 0, \quad
\text{as }  n\to\infty.\label{e3.9}
\end{equation}
By \eqref{e3.2}, \eqref{e3.6}--\eqref{e3.9} we obtain
\begin{equation}
\|v^*\|^{2}=\int_{\Omega}f(v^*(x,0))v^*(x,0)dx.\label{e3.10}
\end{equation}
Thus, \eqref{e3.8} and \eqref{e3.10} give that
$$
\lim_{n\to\infty}\|v_{n}\|^{2}=\|v^*\|^{2}.
$$
Finally we have that
$$
\|v_{n}-v^*\|^{2}=\|v_{n}\|^{2}+\|v^*\|^{2}-2 \langle v_{n},v^*\rangle \to 0 \ \ \text{as} \ n\to\infty.
$$
The proof is complete.
\end{proof}

Next we prove the compactness of the functional 
$\mathcal{J}$ and compute the critical groups of $\mathcal{J}$ 
at infinity. We will use $C_i>0$ to denote various constants independent
of the functions in $H_{0,L}^1(\Omega)$.

\begin{lemma}\label{lem3.2} 
Assume {\rm (A1)} and {\rm (A2)}.
 \begin{itemize}
\item[(i)] The functional $\mathcal{J}$ satisfies the Palais-Smale condition.
\item[(ii)] $C_{q}(\mathcal{J},\infty) \cong0$ for all $q\in \mathbb{Z}$.
\end{itemize}
\end{lemma}

\begin{proof}
 (i) Let $\{v_{n}\}\subset H_{0,L}^{1}(\mathcal{C})$ be such that 
$\{\mathcal{J}(v_{n})\}$ is bounded from above by some $C_1>0$ for all 
$n\in \mathbb{N}$ and
\begin{equation}
\mathcal{J}'(v_n) \to 0\quad \text{in $(H_{0,L}^{1}(\mathcal{C}))^{*}$ as }
 n\to\infty .\label{e3.11}
\end{equation}
 By Lemma \ref{lem3.1} we only need to show that $\{v_n\}$ is bounded in
$H_{0,L}^{1}(\mathcal{C})$. Now it follows from (A1) and (A2)
 that for $n$ large,
\begin{equation}
 \begin{aligned}
 & \theta C_1 + \|v_{n}\| \\ \\
&\geqslant \theta \mathcal{J}(v_{n})-\langle\mathcal{J}'(v_{n}),v_{n}\rangle \\
&= \frac{\theta-2}{2}\|v_{n}\| ^{2}-\int_{\Omega}\Big(\theta F(v_n(x,0))
 -f(v_n(x,0))v_n(x,0)\Big)dx\\
&=  \frac{\theta-2}{2} \|v_{n}\| ^{2}-\int_{\{|v_n(x,0)|\geqslant R\}}
 \Big(\theta F(v_n(x,0))-f(v_n(x,0))v_n(x,0)\Big)dx\\
&\quad  - \int_{\{|v_n(x,0)|<R\}}\Big(\theta F(v_n(x,0))-f(v_n(x,0))v_n(x,0)\Big)dx\\
&\geqslant  \frac{\theta-2}{2} \|v_{n}\| ^{2}
 - \int_{\{|v_n(x,0)|\leqslant R\}}|\theta F(v_n(x,0))
 -f(v_n(x,0))v_n(x,0)|dx\\
&\geqslant  \frac{\theta-2}{2} \|v_{n}\| ^{2}- C_2
 \end{aligned}\label{e3.12}
\end{equation}
where
$$
C_2 = |\Omega|\sup_{|t|\leqslant R}|\theta F(t)+f(t)t|.
$$
 Since $\theta> 2$, it follows from \eqref{e3.12} that
$\{v_{n}\}$ is bounded in $H_{0,L}^{1}(\mathcal{C})$.
By Lemma \ref{lem3.1} one sees that $\{v_n\}$ has a convergent subsequence .

 (ii) Denote $B_1=\{ v \in H_{0,L}^{1}(\mathcal{C}) : \|v\| \leqslant 1\}$.
 By \eqref{e1.4}, we deduce that there is $C_3>0$ such that
\begin{equation}
F(t) \geqslant C_3 |t|^\theta, \quad \text{for all }
|t| \geqslant R.\label{e3.13}
\end{equation}
For $v \in \partial B_1=\{ v \in H_{0,L}^{1}(\mathcal{C}) :
 \|v\| = 1\}$ and $\eta>0$,
 we have
\begin{align*}
 \mathcal{J}(\eta v)
&= \frac{1}{2} \eta^2 \int_\mathcal{C} |\nabla v|^{2}\,dx\,dy
 -\int_{\Omega}F(\eta v(x,0))dx\\
&=  \frac{1}{2} \eta^{2}-  C_{3} \int_{\{|\eta v(x,0)|\geqslant R\}}
 |\eta v(x,0)|^{\theta} dx
 +\int_{|\eta v(x,0)|< R} |F(\eta v(x,0))|dx \\
&\leqslant  \frac{1}{2}\eta^{2}-
  C_3 \int_{\Omega} |\eta v(x, 0)|^{\theta} d x
 + C_3 \int_{\{|\eta v(x,0)|< R\}} |\eta v(x, 0)|^{\theta}dx \\
&\quad + \int_{\{|\eta v(x,0)|< R\}} |F(\eta v(x,0))|dx \\
&\leqslant  \frac{1}{2}\eta^{2}-C_{3} \eta ^{\theta}
 \|\operatorname{tr}_{\Omega} v\|_{L^{\theta}(\Omega)}^{\theta}+C_4
 \end{align*}
where
 $$
C_4=|\Omega|(C_3 R^\theta+\sup_{|t|\leqslant R}|F(t)|).
$$
 Since $\theta>2$, it follows that
\begin{equation}
\lim_{\eta\to +\infty}\mathcal{J}(\eta v)= -\infty.\label{e3.14}
\end{equation}
For $v \in \partial B_1$ and $\eta>0$, by \eqref{e1.4} we have
\begin{align*}
\frac{d}{d\eta} \mathcal{J}(\eta v)
&=  \langle\mathcal{J}'(\eta v),v\rangle \\
&=  \eta \int_{\mathcal{C}} |\nabla v |^{2}\,dx\,dy
 - \int_{\Omega}f(\eta v(x,0)) v(x,0)dx\\
&= \frac{1}{\eta}\Big(2 \mathcal{J}(\eta v)
 + \int_{\Omega}(2F(\eta v(x,0))- f(\eta v(x,0)) \eta v(x,0))dx\Big)\\
&\leqslant  \frac{1}{\eta} \Big(2\mathcal{J}(\eta v)
 +  \int_{\{|\eta v(x,0)|\leqslant R\}}(2F(\eta v(x,0))
 - f(\eta v(x,0)) \eta v(x,0))dx\Big)\\
&\leqslant  \frac1\eta (2\mathcal{J}(\eta v)+C_5),
 \end{align*}
where
 $$
C_5 = |\Omega| \sup_{|t|\leqslant R} (2|F(t)|+R|f(t)|).
$$
Therefore, for any a fixed $a < -C_5/2$,
\begin{equation}
\mathcal{J}(\eta v) \leqslant a \; \Rightarrow \;
 \frac{d}{d\eta} \mathcal{J}(\eta v) <0.\label{e3.15}
\end{equation}
 Since $\mathcal{J}(0)=0$, it follows from \eqref{e3.14} and \eqref{e3.15}
that for any $v \in \partial B_1$, there is a unique $\eta(v)>0$ such that
\begin{equation}
\mathcal{J}(\eta(v) v) = a, \quad
v \in \partial B_1.\label{e3.16}
\end{equation}
By \eqref{e3.16} and the Implicit Function Theorem we have that
 $\eta\in C(\partial B_1, \mathbb{R})$. Now we define
\[
 \pi(v) =  \begin{cases} 1, & \text{if }  \mathcal{J}(v) \leqslant  a, \\
 \|v\|^{-1} \eta(\|v\|^{-1} v), & \text{if }
 \mathcal{J}(v) > a,\; v \ne0.
 \end{cases}
\]
 Then $\pi \in C(H_{0,L}^{1}(\mathcal{C})\setminus \{0\}, \mathbb{\mathbb{R}})$.
 Define the mapping
 $\xi: [0, 1] \times H_{0,L}^{1}(\mathcal{C})\setminus \{0\}
\to H_{0,L}^{1}(\mathcal{C})\setminus \{0\}$ by
 $$
\xi(\sigma, v)=(1-\sigma)v+ \sigma \pi(v)v.
$$
It is easy to see that $\xi$ is continuous. For all
$v\in H_{0,L}^{1}(\mathcal{C})\setminus \{0\}$ with
$\mathcal{J}(v)>a$, by \eqref{e3.16},
 $$
\mathcal{J}(\xi(1, v))=\mathcal{J}(\pi(v)v)
= \mathcal{J}(\eta(\|v\|^{-1}v) \|v\|^{-1} v)=a.
$$
 Therefore
$\xi(1, v) \in \mathcal{J}^a$ for all $v\in H_{0,L}^{1}(\mathcal{C})\setminus \{0\}$,
and $\xi(\sigma, v)= v$ for all $\sigma\in [0, 1]$, $v\in \mathcal{J}^a$.
Then $\mathcal{J}^a$ is a strong deformation retract of
 $H_{0,L}^{1}(\mathcal{C})\setminus \{0\}$.
 It follows that
\begin{align*}
C_{q}(\mathcal{J},\infty)
&= H_{q}(H_{0,L}^{1}(\mathcal{C}),\mathcal{J}^{a}) \\
&\cong H_{q}(H_{0,L}^{1}(\mathcal{C}), H_{0,L}^{1}(\mathcal{C})\setminus \{0\})\\
&\cong  H_{q}( B_1, \partial B_1)\cong 0, \quad  q\in \mathbb{Z},
\end{align*}
 since $\partial B_1$ is contractible which follows from
$\dim H_{0,L}^{1}(\mathcal{C}) =\infty$. The proof is complete.
\end{proof}

 We remark here that the idea for computing $C_q(\mathcal{J}, \infty)\cong 0$ 
is essentially from the famous paper \cite{Wang1991} where 
superlinear Laplacian problems were studied. We use this idea for 
superlinear problems involved with the square root of Laplacian.

\begin{lemma} \label{lem3.3} 
Assume {\rm (A3)}. Then
 \begin{itemize}
 \item[(i)] the functional $\mathcal{J}$ satisfies the Cerami condition.
 \item[(ii)] {\rm (A3)} with $+$ in \eqref{e1.6} implies 
 $C_{\ell_{\infty}}(\mathcal{J},\infty) \not \cong0$, 
where $\ell_{\infty}=\dim H^-(\mu_m)$.

 \item[(iii)] {\rm (A3)} with $-$ in \eqref{e1.6}  implies 
 $C_{\ell^*_\infty}(\mathcal{J},\infty) \not \cong0$, where
 $\ell^*_\infty =\dim \big[H^-(\mu_m) \oplus H(\mu_m)\big]$.
 \end{itemize}
 \end{lemma}

\begin{proof} 
Denote
 $$
g(t)=f(t)-\mu_{m}t, \quad 2G(t)=2F(t)- \mu_{m} t^{2}.
$$ 
We first note that \eqref{e1.5} implies (A1) and
\begin{equation}
\lim_{|t|\to \infty}\frac{2G(t)}{t^{2}}
=\lim _{|t|\to \infty}\frac{g(t)}{t}=0.\label{e3.17}
\end{equation}
By \eqref{e1.6} we have
\begin{equation}
\lim_{|t|\to\infty} \pm(g(t)t-2G(t))
= \lim_{|t|\to\infty} \pm(f(t)t - 2F(t))
=+ \infty.\label{e3.18}
\end{equation}
It follows from \eqref{e3.17} that for any $\epsilon>0$ there exists
$C_\epsilon>0$ such that
\begin{equation}
|g(t)| \leqslant \epsilon |t| +C_\epsilon \quad \text{for all }
 t\in \mathbb{R}.\label{e3.19}
\end{equation}
We rewrite the functional $\mathcal{J}$ defined by \eqref{e2.18} as
\begin{equation}
\mathcal{J}(v)= \frac{1}{2}\int_{\mathcal{C}} |\nabla v|^{2}\,dx\,dy
 - \frac{\mu_{m}}{2}\int_{\Omega}| v(x,0)|^{2}dx
-\int_{\Omega} G( v(x,0))dx, \label{e3.20}
\end{equation}
 and rewrite the derivative of $\mathcal{J}$ as
\begin{equation}
\begin{aligned}
\langle \mathcal{J}'(v), \phi\rangle
&=  \int_{\mathcal{C}} \nabla v\nabla \phi \,dx\,dy
-\mu_{m}\int_{\Omega}v(x,0)\phi(x,0)dx \\
&\quad -\int_{\Omega}g( v(x,0))\phi(x,0)dx .
\end{aligned} \label{e3.21}
\end{equation}


 (i)  Now we begin to satisfy the Cerami condition. 
Let $\{v_n\}\subset H_{0,L}^{1}(\mathcal{C})$ be such that
\begin{gather}
 \mathcal{J}(v_n) \to c \in \mathbb{R} \quad \text{as } 
 n\to\infty\label{e3.22} \\
(1+\|v_n\|)\|\mathcal{J}'(v_n)\|_* \to 0 \quad \text{as }
 n\to\infty.\label{e3.23}
\end{gather}
 We first show that $\{v_n\}$ is bounded in $ H_{0,L}^{1}(\mathcal{C})$. 
By the way of contradiction, we assume that
\begin{equation}
\|v_n\| \to \infty,\ \ \ n \to\infty.\label{e3.24}
\end{equation}
Set $w_n=\frac{v_n}{\|v_n\|}$. Then $\|w_n\|\equiv 1$ for all $n\in \mathbb{N}$.
By Proposition \ref{prop2.1}, up to a subsequence if necessary, there is some $w^*\in
H_{0,L}^{1}(\mathcal{C})$ satisfying
\begin{equation}
\begin{gathered}
 w_n \rightharpoonup w^*, \quad
 \text{weakly in }  \ H_{0,L}^{1}(\mathcal{C})\\
 \operatorname{tr}_{\Omega}w_n\to \operatorname{tr}_{\Omega}w^* \quad
 \text{strongly in }  L^q(\Omega) \quad \forall  q \in [1, 2^{\sharp}),\\
 w_n(x,0)\to w^*(x,0) \quad\text{a.e.  in }  \Omega
 \end{gathered}\label{e3.25}
\end{equation}
 as $n\to \infty$, and there exists $\psi\in L^q(\Omega)$ such that
\begin{equation}
|w_n(x,0)|\leqslant \psi(x)\quad  \text{a.e. in $\Omega$ for any }
 n \in \mathbb{N}.\label{e3.26}
\end{equation}
By \eqref{e3.21} and \eqref{e3.23}, we have that for any
$\phi \in H_{0,L}^{1}(\mathcal{C})$,
\begin{equation}
\begin{aligned}
 \langle \mathcal{J}'(v_{n}), \phi\rangle
&=   \int_{\mathcal{C}} \nabla v_{n}\nabla \phi \,dx\,dy
 -\mu_{m}\int_{\Omega}v_{n}(x,0)\phi(x,0)dx\\
&\quad -\int_{\Omega}g(v_{n}(x,0))\phi(x,0)dx\to 0 \quad \text{as }
 n\to \infty.
\end{aligned}\label{e3.27}
\end{equation}
 Take $\phi=w_n-w^*$ in \eqref{e3.27}, and divide it by $\|v_n\|$, we get
\begin{equation}
\begin{aligned}
&\int_\mathcal{C} \nabla w_n \nabla(w_n-w^*) dx
  - \mu_{m} \int_\Omega w_n(x,0) (w_n(x,0)-w^*(x,0)) dx\\
&- \int_\Omega \frac{g(v_n(x,0))}{\|v_n\|} (w_n(x,0)-w^*(x,0))dx \to0
\quad \text{as }  n\to\infty.
\end{aligned}\label{e3.28}
\end{equation}
By \eqref{e3.19}, \eqref{e3.25}, Proposition \ref{prop2.1} and the H\"older inequality,
we have
\begin{equation}
\begin{aligned}
& \big|\int_\Omega \frac{g(v_n(x,0))}{\|v_n\|} (w_n(x,0)-w^*(x,0))dx\big|\\
&\leqslant   \frac{1}{\|v_n\|} \int_\Omega (\epsilon |v_{n}(x,0)|+ C_\epsilon
)|w_n(x,0)-w^*(x,0)|dx \\
&\leqslant  \epsilon\|\operatorname{tr}_{\Omega}w_n\|_{L^2(\Omega)}
 \|\operatorname{tr}_{\Omega}w_n-\operatorname{tr}_{\Omega}w^*\|_{L^2(\Omega)} \\
&\quad  + \frac{C_\epsilon }{\|v_n\|} \int_\Omega|w_n(x,0)-w^*(x,0)|dx\\
&\leqslant  \epsilon c_{2} \|\operatorname{tr}_{\Omega}w_n
 -\operatorname{tr}_{\Omega}w^*\|_{L^2(\Omega)}
 + C_\epsilon \frac{\|\operatorname{tr}_{\Omega}w_n
 -\operatorname{tr}_{\Omega}w^*\|_{L^1(\Omega)}}{\|v_n\|} \\
&\to 0 \quad \text{as } n\to\infty,
 \end{aligned}\label{e3.29}
\end{equation}
where $c_2$ is the embedding constant of
$H_{0,L}^1(\mathcal{C})\hookrightarrow L^2(\Omega)$.
Moreover, \eqref{e3.25} implies that
\begin{equation}
\int_\Omega w_n(x,0) (w_n(x,0)-w^*(x,0)) dx \to0 \quad \text{as }
n\to\infty.\label{e3.30}
\end{equation}
 It follows from \eqref{e3.28}, \eqref{e3.29} and \eqref{e3.30} that
 $$
\langle w_n, w_n-w^* \rangle=\int_\mathcal{C} \nabla w_n \nabla(w_n-w^*) \,dx\,dy
 \to0 \quad \text{as }  n\to\infty.
$$
 By \eqref{e3.25}, it is clear that
 $$
\langle w^*, w_n-w^* \rangle
=\int_\mathcal{C} \nabla w^* \nabla(w_n-w^*) \,dx\,dy \to 0 \quad
 \text{as }  n\to\infty.
$$
 Thus
 $$
\|w_n-w^*\|^2 = \langle w_n-w^*, w_n-w^*\rangle \to 0 \quad \text{as }
n\to\infty.
$$
This proves
\begin{equation}
w_n \to w^* \quad \text{strongly in }  H_{0,L}^{1}(\mathcal{C})\label{e3.31}
\end{equation}
and $\|w^*\|=1$.
 Now dividing by $\|v_n\|$ in \eqref{e3.27}, we deduce that for all
$ \phi \in H_{0,L}^{1}(\mathcal{C})$,
\begin{equation}
\int_{\mathcal{C}} \nabla w_{n} \nabla \phi \,dx\,dy
-\mu_{m}\int_{\Omega}w_{n}(x,0)\phi(x,0)dx
-  \int_{\Omega}\frac{g(v_{n}(x,0))}{\|v_{n}\|}\phi(x,0)dx \to 0\label{e3.32}
\end{equation}
as $n\to\infty$.  Since for each $\phi\in H_{0,L}^{1}(\mathcal{C})$,
by \eqref{e3.19} we have
\begin{align*}
&\big|\int_{\Omega}\frac{g(v_{n}(x,0))}{\|v_{n}\|}\phi(x,0)dx\big| \\
&\leqslant   \epsilon\|\operatorname{tr}_{\Omega} w_{n}\|_{L^2(\Omega)}
 \|\operatorname{tr}_{\Omega}\phi\|_{L^2(\Omega)}
 + \frac{C_\epsilon\|\operatorname{tr}_{\Omega}\phi\|_{L^1(\Omega)}}{\|v_n\|} \\
&\leqslant  \epsilon c_{2} \|\operatorname{tr}_{\Omega}\phi\|_{L^2(\Omega)}
  + \frac{C_\epsilon\|\operatorname{tr}_{\Omega}\phi\|_{L^1(\Omega)}}{\|v_n\|},
 \end{align*}
 it follows that
\begin{equation}
\lim_{n\to \infty} \int_{\Omega}\frac{g(v_{n}(x,0))}{\|v_{n}\|}\phi(x,0)dx=0,
\quad \forall  \phi\in H_{0,L}^{1}(\mathcal{C}).\label{e3.33}
\end{equation}
 By \eqref{e3.31}, \eqref{e3.32} and \eqref{e3.33}, setting $n\to\infty$,
we have
 $$
\int_{\mathcal{C}} \nabla w^{*} \nabla \phi dx
= \mu_{m}\int_{\Omega} w^{*}(x,0)\phi(x,0)dx, \quad \forall
 \phi \in H_{0,L}^{1}(\mathcal{C}).
$$
 Therefore $w^{*} $ weakly solves the linear elliptic equation in the
cylinder $\mathcal{C}$,
\begin{gather*}
- \Delta w^*=0 \quad  \text{in } \mathcal{C},\\
 w^*=0 \quad  \text{on } \partial _{L}\mathcal{C},\\
  \frac{\partial w^*}{\partial \nu}= \mu_{m} w^*(x,0) \quad
\text{on } \Omega \times \{0\}.
\end{gather*}
This means that $w^{*}(\cdot,0) $ is an eigenfunction corresponding to
the eigenvalue $\mu_{m}$ of the operator $A_{1/2}$ therefore an eigenfunction
corresponding to $\lambda_{m}$ of the operator $-\Delta$.
 By the unique continuity property of the eigenfunctions of $-\Delta$,
we have that
 $w^{*}(x, 0) \ne 0$ a.e in $\Omega$.
 Thus by \eqref{e3.24} and \eqref{e3.25} we obtain
 $$
|v_n(x, 0)|= \|v_n\||w_n(x, 0)| \to \infty \quad \text{uniformly for a.e. }
 x \in \Omega.
$$
 It follows from \eqref{e3.18} that
 $$
\lim_{n\to \infty} \big(f(v_n(x,0)) v_n(x,0)-2F(v_n(x,0))\big)
=\pm\infty \quad \text{uniformly for a.e. }  x \in \Omega.
$$
 Then Fatou's lemma gives
\begin{equation}
\int_\Omega \Big(f(v_n(x,0)) v_n(x,0)-2F(v_n(x,0))\Big) dx \to \pm \infty.
\label{e3.34}
\end{equation}
 On the other hand, it follows from \eqref{e3.22} and \eqref{e3.23} that
 $$
2\mathcal{J}(v_n) - \langle \mathcal{J}'(v_n), v_n \rangle \to 2c,
$$
 therefore
 $$
\int_\Omega \Big(f(v_n(x,0)) v_n(x,0)-2F(v_n(x,0))\Big) dx
= 2\mathcal{J}(v_n) - \langle \mathcal{J}'(v_n), v_n \rangle \to 2c,
$$
which  contradicts \eqref{e3.34}. Thus $\{v_n\}$ is bounded and then
the Cerami condition follows from Lemma \ref{lem3.1}.


 (ii)  We will prove that the functional $\mathcal{J}$ has the geometric feature 
required by Proposition \ref{prop2.4} with respect to the orthogonal splitting 
 (see \eqref{e2.13})
$$
H_{0,L}^{1}(\mathcal{C})= H^-(\mu_m) \oplus \big[H(\mu_m)\oplus H^+(\mu_m)\big]
:= V_\infty \oplus W_\infty.
$$ 
By \eqref{e3.17}, for $\epsilon>0$ small, there is $M_\epsilon>0$ such that
\begin{equation}
G(t) \geqslant -\frac12 \epsilon t^2 - M_\epsilon \quad \text{for all }
 t \in \mathbb{R}.\label{e3.35}
\end{equation}
 Thus
 $$
\int_{\Omega}G(v(x,0))dx \geqslant -\frac12 \epsilon \int_\Omega |v(x,0)|^2 d x
- M_\epsilon |\Omega|.
$$
 For $v \in H^-(\mu_m)$, by \eqref{e3.35} and Propositon \ref{prop2.3},
 we have 
\begin{align*}
 \mathcal{J}(v)
&=  \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy
 - \frac{\mu_{m}}{2}\int_{\Omega}|v(x,0)|^{2}dx
 - \int_{\Omega} G(v(x,0)) dx\\
&\leqslant   \frac12 \int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy
 - \frac12(\mu_m-\epsilon)  \int_{\Omega} |v(x,0)|^2 dx +M_\epsilon |\Omega|
 \\
&\leqslant   \frac12 \Big(1-\frac{\mu_m-\epsilon}{\mu_{m-1}}\Big)
\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy + M_\epsilon |\Omega|
 \end{align*}
 Let $\epsilon \in (0, \mu_m-\mu_{m-1})$ be fixed. Then we have that
 \begin{equation}
\mathcal{J}(v) \to-\infty \quad \text{for $v\in H^-(\mu_m)$ with }
 \|v\|\to \infty.\label{e3.36}
\end{equation}
It follows from \eqref{e3.18} and (A3) with $+$ in \eqref{e1.6},
that for every $T>0$, there is $M>0$ such that
 $$
g(t)t-2G(t) \geqslant T \quad\text{for all }  |t|\geqslant M.
$$
 For $t>0$, we have
\begin{equation}
\frac{d}{dt}\Big[\frac{G(t)}{t^{2}}\Big]=\frac{g(t)t-2G(t)}{t^{3}}.\label{e3.37}
\end{equation}
 Integrating \eqref{e3.37} over $[t, s] \subset [M,  \infty)$, we obtain
 $$
\frac{G(s)}{s^{2}}-\frac{G(t)}{t^{2}}\geqslant \frac{T}{2}
\big(\frac{1}{t^{2}}-\frac{1}{s^{2}}\big).
$$
 Letting $s\to +\infty$ and using \eqref{e3.17}, we see that
 $$
G(t)\leqslant -\frac{T}{2} \quad \text{for } t\geqslant M.
$$
A similar process shows that
$$
G(t)\leqslant -\frac{T}{2} \quad  \text{for all }  t \leqslant -M.
$$
 Hence
\begin{equation}
\lim_{|t|\to \infty} G(t)=-\infty.\label{e3.38}
\end{equation}
 For $v \in H(\mu_m)\oplus H^+(\mu_m)=\big[H^-(\mu_m)\big]^\perp$,
we write $v = {\bar{v}}+\tilde{v}$, $\bar{v} \in H(\mu_m), \tilde{v} \in H^+(\mu_m)$.
Then by Propostion \ref{prop2.3} we have
\begin{align*}
\mathcal{J}(v)
&=  \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy
 - \frac{\mu_{m}}{2}\int_{\Omega}|v(x,0)|^{2}dx
 - \int_{\Omega} G(v(x,0)) dx\\
&= \frac{1}{2}\int_{\mathcal{C}}|\nabla \tilde{v}|^{2}\,dx\,dy
 - \frac{\mu_{m}}{2}\int_{\Omega}|\tilde{v}(x,0)|^{2}dx
 - \int_{\Omega} G(v(x,0)) dx \\
&\geqslant  \frac12\Big(1-\frac{\mu_m}{\mu_{m+\ell_m}}\Big)
  \int_{\mathcal{C}}|\nabla \tilde{v}|^{2}\,dx\,dy
 - \int_{\Omega}G(v(x,0))dx\\
\end{align*}
By \eqref{e3.38}, we see that for some $K>0$ it holds that
$$
G(t) \leqslant K \quad \text{for all } t\in \mathbb{R}.
$$
Therefore
$$
\mathcal{J}(v) \geqslant  \frac12\Big(1-\frac{\mu_m}{\mu_{m+\ell_m}}\Big)
\int_{\mathcal{C}}|\nabla \tilde{v}|^{2}\,dx\,dy - K|\Omega|.
$$
It follows that
\begin{equation}
\mathcal{J}(v) = \mathcal{J}(\tilde{v}+\bar{v}) \to \infty \quad
 \text{as } \|\tilde{v}\| \to \infty.\label{e3.39}
\end{equation}
 Now we show that
\begin{equation}
\|\tilde{v}\|  \text{ bounded and }  \|\bar{v}\| \to \infty
\; \Longrightarrow \; \mathcal{J}(v) = \mathcal{J}(\tilde{v}+\bar{v}) \to \infty.
\label{e3.40}
\end{equation}
 We only need to show that for any $\{v_n=\tilde{v}_n+\bar{v}_n\}$ such that
 $\{\|\tilde{v}_n\|\}$ is bounded and $\|\bar{v}_n\| \to \infty$ implies
$\mathcal{J}(\tilde{v}_n+\bar{v}_n) \to \infty$ as $n\to\infty$.

 Set $w_n=v_n/\|v_n\|$ and write $w_n = \tilde{w}_n + \bar{w}_n$, 
$\tilde{w}_n= \frac{\tilde{v}_n}{\|v_n\|},
 \bar{w}_n = \frac{\bar{v}_n}{\|v_n\|}$. We have
\begin{equation}
\|\tilde{w}_n\|\to 0 \quad \text{and} \quad
 \|\bar{w}_n\|\to 1 \quad \text{as }  n\to\infty.\label{e3.41}
\end{equation}
 By Proposition \ref{prop2.1}, up to a subsequence if necessary, there is some
 $w_* \in H(\mu_m)\oplus H^+(\mu_m)$ such that
\begin{equation}
\begin{gathered}
 w_n \rightharpoonup w_*, \quad \text{weakly in } H_{0,L}^{1}(\mathcal{C})\\
 \operatorname{tr}_{\Omega} w_n\to \operatorname{tr}_{\Omega}w_* \quad
 \text{strongly in }  L^2(\Omega),\\
 w_n(x,0)\to w_*(x, 0) \quad \text{ a.e.  in }  \Omega
 \end{gathered}\label{e3.42}
\end{equation}
 as $n\to \infty$. From \eqref{e3.41} and \eqref{e3.42} we deduce that
$w_*\in E(\mu_m)$ and $\|w_*\|=1$.
 Therefore $w_*$ weakly solves the linear elliptic equation in the cylinder
$\mathcal{C}$,
\begin{gather*}
-\Delta w_*=0  \quad \text{in } \mathcal{C},\\
 w_*=0 \quad  \text{on } \partial _{L}\mathcal{C},\\
  \frac{\partial w}{\partial \nu}= \mu_{m}w_*(x,0)
 \quad \text{on } \Omega \times \{0\}.
\end{gather*}
This means that $w_*(\cdot, 0)$ is an eigenfunction corresponding to the
eigenvalue $\mu_m$ of $A_{1/2}$. Therefore
 $w(x, 0) \ne0$ for a.e. $x\in \Omega$.
 It follows that
\begin{equation}
|v_n(x, 0)| = \|v_n\||v_n(x, 0)| \to\infty, \quad \text{for a.e.
 $x\in \Omega$, as }  n\to\infty.\label{e3.43}
\end{equation}
 Now by \eqref{e3.38}, \eqref{e3.43} and Fatou's lemma, we obtain
 $$
\mathcal{J} (v_n) \geqslant  \frac12\Big(1-\frac{\mu_m}{\mu_{m+\ell_m}}\Big)
 \int_{\mathcal{C}}|\nabla \tilde{v}_n|^{2}\,dx\,dy
- \int_\Omega G(v_n(x, 0))dx \to \infty
$$
as $n\to\infty$. This proves \eqref{e3.40}. Now \eqref{e3.39} and \eqref{e3.40}
imply
\begin{equation}
\mathcal{J}(v)\to\infty \quad \text{for $ v \in H(\mu_m)\oplus H^+(\mu_m)$ with }
\|v\|\to\infty.\label{e3.44}
\end{equation}
 It follows from the fact of $\mathcal{J}$ being weakly lower semicontinuous
on $H(\mu_m)\oplus H^+(\mu_m)$ and \eqref{e3.44} that $\mathcal{J}$ is bounded
from below on $H(\mu_m)\oplus H^+(\mu_m)$. Finally by Proposition \ref{prop2.4}
we get
 $C_{\ell_\infty}(J,\infty)\not\cong0$, where $\ell_\infty=\dim H^-(\mu_m)$.


(iii) In a similar way we can prove that the functional
 $\mathcal{J}$ has the geometric feature required by
 Proposition \ref{prop2.4} with respect to the orthogonal splitting  
(see \eqref{e2.13})
 $$
H_{0,L}^{1}(\mathcal{C})=\big[H^-(\mu_m) \oplus H(\mu_m)\big] 
\oplus H^+(\mu_m):= V_\infty \oplus W_\infty.
$$
 Therefore,
 $C_{\ell^*_\infty}(J,\infty)\not\cong0$, where 
$\ell^*_\infty=\dim H^-(\mu_m)\oplus H(\mu_m)$.
 The proof is complete. 
\end{proof}

\begin{lemma}\label{lem3.4} 
Assume {\rm (A1)} and {\rm (A4)}.
 \begin{itemize}
 \item[(i)] The functional $\mathcal{J}$ is coercive on 
 $H^1_{0, L}(\mathcal{\mathcal{C}})$.
 \item[(ii)] The functional $\mathcal{J}$ satisfies the Palais-Smale condition.
\item[(iii)] $C_{q}(\mathcal{J},\infty) \cong \delta _{q,0} \mathbb{Z}$.
\end{itemize}
\end{lemma}

\begin{proof}
 (i) For $v \in H_{0,L}^{1}(\mathcal{C})$, we have by (A4) that
\begin{align*}
 \mathcal{J}(v) 
&= \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy-\int_{\Omega}F(v(x,0))dx \\
&\geqslant  \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy
 -\frac{1}{2}\mu\int_{\Omega}|v(x,0)|^{2}dx-C|\Omega| \\
&\geqslant  \frac{1}{2}\Big(1-\frac{\mu}{\mu_{1}}\Big)\|v\|^{2}
 - C|\Omega|.
 \end{align*} 
 Since $\mu<\mu_1$, we have that
 $\mathcal{J}(v) \to \infty$ as $\|v\|\to \infty$.
 This proves that $\mathcal{J}$ is coercive.

 (ii) Let $\{v_n\} \subset H_{0,L}^1(\mathcal{C})$ be a Palais-Smale 
sequence at $c\in \mathbb{\mathbb{R}}$. By the coerciveness of 
$\mathcal{J}$, $\{v_n\}$ is bounded and then by Lemma \ref{lem3.1} 
it contains convergent subsequence.

 (iii) Since $\mathcal{J}$ is coercive and is weakly lower semicontinuous 
on $H^1_{0, L}(\mathcal{\mathcal{C}})$,
 $\mathcal{J}$ attains its global minima $\inf \mathcal{J}$ at some $v_*$:
 $$
\mathcal{J}(v_*)=\min_{v\in H^1_{0, L}(\mathcal{\mathcal{C}})} \mathcal{J}(v).
$$ 
Take $a< \mathcal{J}(v_*)$. Then
 $$
C_q(\mathcal{J}, \infty)=H_q(H^1_{0, L}(\mathcal{\mathcal{C}}), 
\mathcal{J}^a) \cong H_q(\{v_*\}, \emptyset) \cong \delta _{q,0} \mathbb{Z}.
$$ 
The proof is complete. 
\end{proof}


 \begin{lemma}\label{lem3.5} 
Assume {\rm (A5)}. Then
 \begin{itemize}
 \item[(i)] the functional $\mathcal{J}$ is coercive on 
$H^1_{0, L}(\mathcal{\mathcal{C}})$;
 \item[(ii)] the functional $\mathcal{J}$ satisfies the Palais-Smale condition;
\item[(iii)] $C_{q}(\mathcal{J},\infty) \cong \delta _{q,0} \mathbb{Z}$.
\end{itemize}
\end{lemma}


\begin{proof}
(i) We first prove that under the condition (A5) the functional 
$\mathcal{J}$ is coercive on $H_{0,L}^{1}(\mathcal{C})$. 
Denote $2G(t)= 2F(t)-\mu _{1}t^{2}$. Then \eqref{e1.9} implies
\begin{equation}
\lim_{|t|\to \infty}G(t)=-\infty.\label{e3.45}
\end{equation}
Rewrite $\mathcal{J}$ as
$$
\mathcal{J}(v)=\frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy
- \frac{\mu_{1}}{2}\int_{\Omega}|v(x,0)|^{2}dx
- \int_{\Omega}G(v(x,0))dx,
$$
for $v\in H_{0,L}^{1}(\mathcal{C})$.
Assume that $\mathcal{J}$ is not coercive on $H_{0,L}^{1}(\mathcal{C})$,
then there is a sequence $\{v_{n}\}\subset H_{0,L}^{1}(\mathcal{C})$ such that
\begin{equation}
\|v_{n}\|\to \infty \quad \text{as } n\to \infty\label{e3.46}
\end{equation}
and
\begin{equation}
\mathcal{J}(v_{n}) \leqslant C \quad  \text{for all } n\in \mathbb{N}.\label{e3.47}
\end{equation}
 for some $C \in \mathbb{R}$.
 Set $w_n=\frac{v_n}{\|v_n\|}$ then $\|w_n\|\equiv 1$ for all $n\in \mathbb{N}$.
 By Proposition \ref{prop2.1}, up to a subsequence if necessary, there is a
$w^*\in H_{0,L}^{1}(\mathcal{C})$ satisfying
\begin{equation}
\begin{gathered}
 w_n \rightharpoonup w^*\quad  \text{weakly in }  H_{0,L}^{1}(\mathcal{C}),\\
 \operatorname{tr}_{\Omega}w_n\to \operatorname{tr}_{\Omega}w^* \quad
 \text{strongly in }  L^2(\Omega),\\
 w_n(x,0)\to w^*(x,0) \quad  \text{a.e. in } \Omega.
 \end{gathered}\label{e3.48}
\end{equation}
 By \eqref{e3.45} we see that $G(t)$ is bounded from above by some constant
$K>0$ for all $t\in \mathbb{R}$.
 Now from \eqref{e3.47} we deduce that
\begin{equation}
\frac C{\|v_{n}\|^{2}} \geqslant\frac{\mathcal{J}(v_{n})}{\|v_{n}\|^{2}}
\geqslant \frac{1}{2}\int_{\mathcal{C}}|\nabla w_{n}|^{2}\,dx\,dy
- \frac{\mu_{1}}{2}\int_{\Omega}|w_{n}(x,0)|^{2}dx
-\frac{K|\Omega|}{\|v_{n}\|^{2}}.\label{e3.49}
\end{equation}
 It follows from \eqref{e3.46}, \eqref{e3.48} and \eqref{e3.49} that
\begin{equation}
\limsup_{n\to \infty}\int_{\mathcal{C}}|\nabla w_{n}|^{2}\,dx\,dy
\leqslant \mu_{1}\int_{\Omega}|w^{*}(x,0)|^{2}dx.\label{e3.50}
\end{equation}
On the other hand,  by the variational characterization of $\mu_{1}$
and the lower semicontinuity of the norm, we have
\begin{equation}
\mu_{1}\int_{\Omega}|w^{*}(x,0)|^{2}dx
\leqslant\int_{\mathcal{C}}|\nabla w^{*}|^{2}\,dx\,dy
\leqslant\liminf_{n\to \infty}\int_{\mathcal{C}}|\nabla w_{n}|^{2}\,dx\,dy.
\label{e3.51}
\end{equation}
From \eqref{e3.50} and \eqref{e3.51} we have that
\begin{gather}
 \lim_{n\to\infty} \|w_n\|^2 = \|w^*\|^2\label{e3.52}, \\
\int_{\mathcal{C}}|\nabla w^{*}|^{2}\,dx\,dy
 =\mu_{1} \int_{\Omega}|w^{*}(x,0)|^{2}dx.\label{e3.53}
\end{gather}
Since $H_{0,L}^{1}(\mathcal{C})$ is a Hilbert space, we have by \eqref{e3.48}
and \eqref{e3.52} that
$$
w_{n} \to w^* \quad \text{strongly in $H_{0,L}^{1}(\mathcal{C})$ as } n\to\infty.
$$
Hence $\|w^{*}\|=1$ and by \eqref{e3.53} we see that
$ w^{*}(x,0)= \pm (\mu_{1})^{-\frac{1}{2}} \varphi_{1}(x)$.
This implies
\begin{equation}
|v_{n}(x,0)|\to \infty \quad \text{uniformly for a.e. }
 x \in \Omega.\label{e3.54}
\end{equation}
Now by \eqref{e3.45}, \eqref{e3.47}, \eqref{e3.54} and the Fatou's lemma
we have that
\begin{align*}
{C}
& \geqslant \frac{1}{2}\int_{\mathcal{C}}|\nabla v_{n}|^{2}\,dx\,dy
 - \frac{\mu_{1}}{2}\int_{\Omega}|v_{n}(x,0)|^{2}dx
 - \int_{\Omega}G(v_{n}(x,0))dx\\
& \geqslant - \int_{\Omega}G(v_{n}(x,0))dx \to \infty \quad \text{as }
n \to \infty.
 \end{align*}
This contradiction shows that $\mathcal{J}$ is coercive on
$H_{0,L}^{1}(\mathcal{C})$.

 (ii) Let $\{v_n\} \subset H_{0,L}^1(\mathcal{C})$ be a Palais-Smale 
sequence at $c\in \mathbb{\mathbb{R}}$. By the coerciveness of 
$\mathcal{J}$, $\{v_n\}$ is bounded and then by Lemma \ref{lem3.1} it 
contains a convergent subsequence.

 (iii) Since $\mathcal{J}$ is coercive and is weakly lower semicontinuous
 on $H^1_{0, L}(\mathcal{\mathcal{C}})$,
 $\mathcal{J}$ attains its global minima $\inf \mathcal{J}$ at some $v_*$:
 $$
\mathcal{J}(v_*)=\min_{v\in H^1_{0, L}(\mathcal{\mathcal{C}})} \mathcal{J}(v).
$$ 
Take $a< \mathcal{J}(v_*)$. Then
 $$
C_q(\mathcal{J}, \infty)=H_q(H^1_{0, L}(\mathcal{\mathcal{C}}), \mathcal{J}^a) 
\cong H_q(\{v_*\}, \emptyset) \cong \delta _{q,0} \mathbb{Z}.
$$ 
The proof is complete. 
\end{proof}

 \section{Critical groups at zero}

 In this section we compute the critical groups of the functional $\mathcal{J}$ 
at zero. We will use $C_i>0$ to denote various constants independent of the 
functions in $H_{0,L}^1(\mathcal{C})$.
 We also make a convention that problem \eqref{eP} has finitely many weak 
solutions and so the trivial solution is an isolated critical point 
of $\mathcal{J}$.

 \begin{lemma} \label{lem4.1} 
Assume {\rm (A1)} and {\rm (A6)}. Then $C_{q}(\mathcal{J},0) \cong 0$ 
for all $q \in \mathbb{Z}$.
 \end{lemma}

\begin{proof}
 By the definition of critical groups, we write
$$
C_{q}(\mathcal{J},0):=H_{q}(B_{\rho}(0)
\cap\mathcal{J}^{0},(B_{\rho}(0)\cap\mathcal{J}^{0})\setminus\{0\}),
$$
where $B_{\rho}(0)=\{v\in H_{0,L}^{1}(\mathcal{C}): \|v\|\leqslant \rho\}$, 
and $\rho>0$ is to be chosen suitable for use. We will construct a 
deformation mapping for the topological pairs 
$(B_{\rho}(0), B_{\rho}(0)\setminus \{0\})$ and 
$(B_{\rho}(0)\cap\mathcal{J}^{0}, (B_{\rho}(0)\cap\mathcal{J}^{0})\setminus\{0\})$.

 A direct calculation by using \eqref{e1.10} and \eqref{e1.11} shows that 
there exists a constant $C_{1}>0$ such that
$$
F(t)\geqslant C_{1} |t|^{\tau} \quad \text{for all }  |t| \leqslant \delta.
$$
By (A1), there exists a constant $C_{2}>0$ such that for some 
$\max\{2, p\} < \gamma < 2^{\sharp}$,
\begin{equation}
|F(t)|\leqslant C_{2}|t|^{\gamma }, \quad  |f(t)t| \leqslant C_2 |t|^\gamma
\quad \text{for all } |t| > \delta.
\label{e4.1}
\end{equation}
 Therefore,
$$
F(t)\geqslant C_{1} |t|^{\tau} - C_{2}|t|^{\gamma } \quad\text{for all }
t \in \mathbb{R}.
$$
Take a function $v \in H_{0,L}^{1}(\mathcal{C})$ with $v\neq 0$,
then for $\eta>0$ we have
\begin{equation}
 \begin{aligned}
 \mathcal{J}(\eta v)
&=  \frac{1}{2}\int_{\mathcal{C}} |\nabla (\eta v)|^{2}\,dx\,dy
 -\int_{\Omega}F(\eta v(x,0))dx\\
&\leqslant \frac{1}{2}\eta^{2}\|v\|^{2}- C_1 \int_{\Omega} |\eta v(x,0)|^{\tau}dx
 + C_2\int_{\Omega} |\eta v(x,0)|^{\gamma}dx \\
&\leqslant \frac{1}{2} \eta^{2}\|v\|^{2}
 - C_{1}\eta^{\tau}\|\operatorname{tr}_{\Omega}v\|_{L^{\tau}(\Omega)}^{\tau}
 +C_{2}\eta^{\gamma} \|\operatorname{tr}_{\Omega}v\|^{\gamma}_{L^{\gamma}(\Omega)}.
 \end{aligned}\label{e4.2}
\end{equation}
Since $1<\tau<2<\gamma< 2^{\sharp}$, one sees from \eqref{e4.2} that for given
$v\in H_{0,L}^{1}(\mathcal{C})$ with $v\neq 0$, there exists
 $\eta_{0}=\eta_{0}(v)>0$ such that
\begin{equation}
\mathcal{J}(\eta v)<0 \quad \text{for all }  0<\eta <\eta_{0}.\label{e4.3}
\end{equation}
 Let $v\in H_{0,L}^{1}(\mathcal{C})$ be such that
 $$
\mathcal{J}(v)=\frac12\int_\mathcal{C} |\nabla v|^2\,dx\,dy
- \int_{\Omega} F(v(x, 0)) dx =0.
$$
It follows from \eqref{e1.11}, (A1) and the continuous embedding
$H_{0,L}^{1}(\mathcal{C})\hookrightarrow L^{q}(\Omega)$ for any
 $q\in [1,2^\sharp]$ that
\begin{align*}
\frac{d}{d\eta}\mathcal{J}(\eta v)\big|_{\eta=1}
&= \int_{\mathcal{C}}|\nabla v |^{2}\,dx\,dy- \int_{\Omega}f(v(x,0)) v(x,0)dx\\
&= \frac{2-\tau}{2} \|v\|^{2}+ \int_{\Omega}\big(\tau F( v(x,0))
 -f( v(x,0)) v(x,0)\big)dx\\
&\geqslant \frac{2-\tau}{2}\|v\|^{2}+ \int_{\{|v(x,0)|>\delta\}}
 \big(\tau F( v(x,0))-f( v(x,0)) v(x,0)\big)dx\\
&\geqslant \frac{2-\tau}{2}\|v\|^{2}- \int_{\{|v(x,0)|>\delta\}}
 \big(|2 F( v(x,0))|+|f( v(x,0) v(x,0)|\big)dx\\
&\geqslant \frac{2-\tau}{2}\|v\|^{2}- C_{3} \int_{\{|v(x,0)|>\delta\}}
 |v(x,0)|^{\gamma}dx \\
&\geqslant \frac{2-\tau}{2}\|v\|^{2}- C_{4} \|v\|^{\gamma}
 \end{align*}
 Thus we can find some $\rho>0$ such that
\begin{equation}
\frac{d}{d\eta}\mathcal{J}(\eta v)\big|_{\eta=1}>0, \quad \text{for }
 v\in H_{0,L}^{1}(\mathcal{C})  \text{ with }  \mathcal{J}(v)=0\quad
 \text{and} \quad  0<\|v\|\leqslant \rho.\label{e4.4}
\end{equation}
By \eqref{e4.3} and \eqref{e4.4}, one sees that for each
$v\in B_{\rho}(0)\setminus \{0\}$ with $\mathcal{J}(v)>0$,
there exists a unique $\eta_{0}=\eta_{0}(v)>0$ such that
\begin{equation}
\mathcal{J}(\eta v)<0\quad \text{for all }  0<\eta<\eta_{0}.\label{e4.5}
\end{equation}
From now on we fix $\rho>0$. We claim that if $v\in B_{\rho}(0)\setminus \{0\}$
and $\mathcal{J}(v)<0$ then
\begin{equation}
\mathcal{J}(\eta v)<0 \ \ \text{for all} \ \eta\in (0, 1).\label{e4.6}
\end{equation}
Let $v\in B_{\rho}(0)$ and $\mathcal{J}(v)<0$. By the continuity of
$\mathcal{J}$, there exists $\vartheta\in (0, 1]$ such that
$$
\mathcal{J}(\eta v)<0 \quad \text{for all }  \eta \in (1-\vartheta, 1).
$$
We show \eqref{e4.6} by proving $\vartheta= 1$.
Suppose that there is some $\eta^{*}\in (0,1-\vartheta]$ such that
$$
\mathcal{J}(\eta^{*}v)=0, \quad \mathcal{J}(\eta v)<0 \quad \text{for all }
 \eta \in (\eta^{*}, \ 1).
$$
Denote $v^{*}=\eta^{*}v$. Then by \eqref{e4.4}, we have
\begin{equation}
\frac{d}{d\eta}\mathcal{J}(\eta v^{*})\big|_{\eta=1}>0.\label{e4.7}
\end{equation}
But $\eta>\eta^{*}$ implies
$$
\mathcal{J}(\eta v)-\mathcal{J}(\eta^{*}v)<0,
$$
which implies
$$
\frac{d}{d\eta}\mathcal{J}(\eta v^{*})\big|_{\eta=1}
=\lim_{\eta\to \eta_{+}^{*}}\frac{\mathcal{J}(\eta v)
-\mathcal{J}(\eta v^{*})}{\eta-\eta^{*}}\leqslant 0.
$$
This contradicts \eqref{e4.7}. Hence $\vartheta=1$ and \eqref{e4.6} holds.

Now we define a mapping $\eta:B_{\rho}(0)\to [0,1]$ by
$$
\eta(v)=  \begin{cases}
 1, & \text{for } v\in B_{\rho}(0) \text{ with } \mathcal{J}(v)\leqslant0, \\
 \eta , & \text{for } v\in B_{\rho}(0) \text{ with } \mathcal{J}(v)>0, 
\mathcal{J}(\eta v)=0, \eta <1.
 \end{cases}
 $$
 By \eqref{e4.4}, \eqref{e4.5} and \eqref{e4.6}, the mapping $\eta$ 
is well-defined and if $\mathcal{J}(v)>0$ then there exists a unique 
$\eta(v)\in (0,1)$ such that
\begin{equation}
\begin{gathered}
 \mathcal{J}(\eta(v)v)=0, \\
 \mathcal{J}(\eta v)<0, \quad \forall  \eta\in (0,\eta(v))\\
 \mathcal{J}(\eta v)>0, \quad \forall  \eta\in (\eta(v),1)\\
 \end{gathered} \label{e4.8}
\end{equation}
 It follows from \eqref{e4.4}, \eqref{e4.8} and the Implicit Function
Theorem that the mapping $\eta$ is continuous in $v$. Define a mapping
$h:[0,1]\times B_{\rho}(0)\to B_{\rho}(0)$ by
 $$
h(t,v)=(1-t)v+t\eta(v)v,\ t\in[0,1],\ v\in B_{\rho}(0).
$$
 It is easy to see that the mapping $h$ is a continuous deformation from
 $(B_{\rho}(0), B_{\rho}(0)\setminus \{0\})$ to
$(B_{\rho}(0)\cap\mathcal{J}^{0}, (B_{\rho}(0)\cap\mathcal{J}^{0})\setminus\{0\})$.
 By the homotopy invariance of homology group, we have for all
$q\in \mathbb{\mathbb{Z}}$,
 $$
C_{q}(\mathcal{J},0)=H_{q}(B_{\rho}(0)\cap\mathcal{J}^{0},
(B_{\rho}(0)\cap\mathcal{J}^{0})\setminus\{0\}) \cong H_{q}(B_{\rho}(0),
B_{\rho}(0)\setminus\{0\})\cong 0.
$$
 since $B_{\rho}(0)\setminus\{0\}$ is contractible. The proof is complete.
\end{proof}

 We remark that the idea for computing critical groups at zero is 
essentially from \cite{liuwu1997} where Laplacian equations with superlinear 
at zero was studied. The similar idea was presented in \cite{moroz1997} 
to deal with also the same problem as in \cite{liuwu1997} using a global 
sign condition $2F(t)-f(t)t>0$ for all $t\ne0$. 
In \cite{jiu-su2003} this idea was used for studying $p$-Laplacian problems.

\begin{lemma}\label{lem4.2} 
Assume that {\rm (A1)} and {\rm (A7)} hold. 
Then $C_{\ell_{0}}(\mathcal{J},0) \not\cong 0$ where
$\ell_0=\dim H^-(\mu_{k+1})$.
 \end{lemma}

\begin{proof}
 We will prove that the functional $\mathcal{J}$ has a local linking structure
at $0$ with respect to the orthogonal splitting  (see \eqref{e2.13})
$$
H_{0,L}^{1}(\mathcal{C})= H^-(\mu_{k+1}) \oplus 
\big[H(\mu_{k+1})\oplus H^+(\mu_{k+1})\big]:= V_0 \oplus W_0.
$$

 (i) For $v \in H^-(\mu_{k+1})$, by Proposition \ref{prop2.1} we have
 $$
\|\operatorname{tr}_{\Omega}v\|_{L^{2}(\Omega)}\leqslant C_{2}\|v\|.
$$
 Note that $\operatorname{tr}_{\Omega}(H^-(\mu_{k+1}))
=\operatorname{span} \{\varphi_{1},\dots, \varphi_{k}\}\subset L^{\infty }(\Omega)$ 
is finite dimensional and all norms are equivalent, we can find a positive 
constant $\rho>0$ such that
$$
\|v\|\leqslant \rho\Rightarrow | v(x,0)|
\leqslant \|\operatorname{tr}_{\Omega}v\|_{L^{\infty}(\Omega)}\leqslant \delta.
$$
It follows from \eqref{e1.12} and Propostion \ref{prop2.3} that for any 
$v\in H^-(\mu_{k+1})$ with $\|v\|\leqslant \rho$, we have
\begin{equation}
\begin{aligned}
 \mathcal{J}(v)
&= \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy
 -\int_{\Omega}F( v(x,0))dx\\
&\leqslant  \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy
- \frac{\mu_{k}}{2}\int_{\Omega}| v(x,0)|^{2}dx \leqslant 0.
 \end{aligned} \label{e4.9}
\end{equation}

 (ii) For $v\in \big[H(\mu_{k+1})\oplus H^+(\mu_{k+1})\big]
= \overline{\operatorname{span} \{e_{k+1},\dots\}}$,
 we write $v = \bar{v} + \tilde{v}$, where
$\bar{v} \in H(\mu_{k+1})$, $\tilde{v}\in H^+(\mu_{k+1})$.
 Then by Propostion \ref{prop2.3} we have
\begin{equation}
\begin{aligned}
 \mathcal{J}(v)
&= \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy-\int_{\Omega}F( v(x,0))dx
 \\
&=  \frac{1}{2}\int_{\mathcal{C}}|\nabla \tilde{v}|^{2}\,dx\,dy
 - \frac{\mu_{k+1}}{2}\int_{\Omega}| \tilde{v}(x,0)|^{2}dx \\
& \quad +\int_{\Omega}\Big(\frac{\mu_{k+1}}{2}| v(x,0)|^{2}-F( v(x,0))\Big)dx\\
&\geqslant  \frac{1}{2}\Big(1-\frac{\mu_{k+1}}{\mu_{k+1+\ell_{k+1}}}\Big)
 \|\tilde{v} \|^{2}+\int_{\Omega}\Big(\frac{\mu_{k+1}}{2}| v(x,0)|^{2}
 -F( v(x,0))\Big)dx.
 \end{aligned}\label{e4.10}
\end{equation}
For $|v(x,0)|\leqslant \delta$, by \eqref{e1.12} we get that
\begin{equation}
\int_{\{|v(x,0)|\leqslant \delta\}} \Big(\frac{1}{2}\mu_{k+1}|
v(x,0)|^{2}-F( v(x,0))\Big)dx\geqslant 0.\label{e4.11}
\end{equation}
 Since $\operatorname{tr}_{\Omega}H(\mu_{k+1})$ is finite dimensional,
there exists $\rho>0$ such that
 $$
\|v\| \leqslant \rho \ \Rightarrow \
\|\operatorname{tr}_{\Omega}\bar{v}\|_{L^{\infty}(\Omega)}
\leqslant \frac{1}{3}\delta.
$$
 For $\|v\| \leqslant \rho$ and $|v(x,0)|> \delta$,
 $$
|\tilde{v}(x,0)| \geqslant |v(x,0)|-|\bar{v}(x,0)|>\frac{2}{3}|v(x,0)|.
$$
 By (A1), take $\max\{2, p\} < \gamma <2^\sharp$, there is $ C_{5}>0$ such that
 $$
\big|\frac12\mu_{k+1} t^{2}-F(t)\big| \leqslant C_{5} |t|^{\gamma} \quad
\text{for all }  |t|\geqslant \delta.
$$
 By Proposition \ref{prop2.1}, we have
\begin{equation}
\begin{aligned}
 & \int_{\{| v(x,0)|> \delta\}}\big|\frac12\mu_{k+1}|v(x,0)|^{2}-F(v(x,0))\big|dx \\
&\leqslant  C_{5} \int_{\{| v(x,0)|> \delta\}}| v(x,0)|^{\gamma}dx \\
&\leqslant  C_{5}(3/2)^{\gamma }  \int_{\Omega}| \tilde{v} (x,0)|^{\gamma }dx\\
&\leqslant  C_{5} (3/2)^{\gamma } C_{\gamma }^{\gamma } \|\tilde{v}\|^{\gamma}
:=C_6\|\tilde{v}\|^{\gamma }.
 \end{aligned}\label{e4.12}
\end{equation}
 Then by \eqref{e4.10}, \eqref{e4.11} and \eqref{e4.12} we get
\begin{equation}
 \mathcal{J}(v) \geqslant \frac{1}{2}
\Big(1-\frac{\mu_{k+1}}{\mu_{k+1+\ell_{k+1}}}\Big)\|\tilde{v}\|^{2}
 -C_6 \|\tilde{v}\|^{\gamma}.\label{e4.13}
\end{equation}
Since $\gamma >2$, we see from \eqref{e4.13} that for $\rho>0$ small
\begin{equation}
\mathcal{J}(v)>0 \quad \text{for } \|v\|\leqslant \rho  \text{ and }
 \tilde{v} \ne0.\label{e4.14}
\end{equation}
 On the other hand, we conclude that for $\|v\| \leqslant \rho $ with
$\tilde{v}=0$ and $\bar{v}\ne0$,
\begin{equation}
\mathcal{J}(v)=\mathcal{J}(\bar{v})
=\int_{\Omega}\Big(\frac{1}{2}\mu_{k+1} \bar{v}^2(x,0)-F(\bar{v}(x,0)) \Big)dx>0.
\label{e4.15}
\end{equation}
Otherwise, if for some $\bar v_* \ne 0 $ and $\|\bar v_*\| \leqslant \rho$
such that $\mathcal{J}(\bar{v}_*)=0$, then by (A7) we have
\begin{equation}
F(\bar{v}_*(x,0)) = \frac{1}{2}\mu_{k+1} \bar{v}_*^2 (x,0) \quad \text{for a.e.}
 x\in \Omega.\label{e4.16}
\end{equation}
 As $v_* \in H(\mu_{k+1})$, all $\sigma v_*$ for $\sigma\in [-1, 1]$
are critical points of $\mathcal{J}$ and so $0$ is not isolated.
It is a contradiction. Therefore we get the conclusion that
\begin{equation}
\mathcal{J}(v)>0 \quad \text{for }
 v \in H(\mu_{k+1})\oplus H^+(\mu_{k+1})  \text{ with }
 0<\|v\|\leqslant \rho.\label{e4.17}
\end{equation}
Now by \eqref{e4.9}, \eqref{e4.17} and Proposition \ref{prop2.5},
 $C_{\ell_0}(J, 0) \not\cong 0$, where $\ell_0=\dim H^-(\mu_{k+1})$.
The proof is complete.
\end{proof}

 \begin{lemma}\label{lem4.3} 
Assume {\rm (A1)} and {\rm (A8)}. Then we have 
$$
C_{q}(\mathcal{J},0) \cong \delta_{q, 0} \mathbb{Z}, \quad q \in \mathbb{Z}.
$$
 \end{lemma}

\begin{proof}
 We will show that $0$ is a strictly local minimizer of $\mathcal{J}$.
 For $v\in H_{0,L}^{1}(\mathcal{C})$, we $v=\bar{v}+\tilde{v}$ where 
$\bar{v} \in H(\mu_1)$ and $\tilde{v} \in H^+(\mu_1)$.
 Take $\rho>0$ small such that
 $$
\|v\| \leqslant \rho \; \Rightarrow \; 
 \|\operatorname{tr}_\Omega \bar v\|_\infty \leqslant \frac13 \delta.
$$ 
Then
\begin{equation}
|v(x,0)|> \delta \;  \Rightarrow \;  |v(x,0)|< \frac{3}{2}|
\tilde{v}(x, 0)|.\label{e4.18}
\end{equation}
 By (A1), take $ \max\{2, p\}<\gamma < 2^\sharp$, there is $C_7>0$ such that
\begin{equation}
\frac12\mu_1 t^2 +|F(t)| \leqslant C_7 |t|^{\gamma } \quad\text{for }
 |t|>\delta.\label{e4.19}
\end{equation}
 By \eqref{e4.18}, \eqref{e4.19} and Proposition \ref{prop2.1}, we have
\begin{equation}
\int_{\{| v(x,0)|> \delta\}}\big|\frac12\mu_{1}|v(x,0)|^{2}-F(v(x,0))\big|dx
\leqslant C_8 \|\tilde{v}\|^{\gamma }.
\label{e4.20}
\end{equation}
Now for $\|v\| \leqslant \rho$, by (A8), \eqref{e4.20} and \eqref{e2.16}, we have
\begin{equation}
\begin{aligned}
 \mathcal{J}(v)
&= \frac{1}{2}\int_{\mathcal{C}}|\nabla v|^{2}\,dx\,dy-\int_{\Omega}F( v(x,0))dx\\
&= \frac{1}{2}\|\tilde{v}\|^{2} -  \frac{\mu_{1}}{2}\int_{\Omega}|
  \tilde{v}(x,0)|^{2}dx \\
&\quad  + \int_{\{| v(x,0)|\leqslant \delta\}}\Big(\frac{1}{2}\mu_{1}| v(x,0)|^{2}
 -F( v(x,0))\Big)dx\\
&\quad + \int_{\{|v(x,0)|>\delta\}}\Big(\frac{\mu_{1}}{2}| v(x,0)|^{2}
 -F( v(x,0))\Big)dx\\
&\geqslant  \frac{1}{2}\Big(1-\frac{\mu_{1}}{\mu_{2}}\Big)\|\tilde{v} \|^{2}
 -C_8 \|\tilde{v}\|^{\gamma }.
 \end{aligned}\label{e4.21}
\end{equation}
 Arguing in the same way as that in the proof of Lemma \ref{lem4.2}
we can prove that $v=0$ is a strictly local minimizer of $\mathcal{J}$.
Thus $C_{q}(\mathcal{J},0) \cong\delta_{q, 0} \mathbb{Z}$, $q\in \mathbb{Z}$.
 The proof is complete.
\end{proof}

 We remark here that (A8) includes the nonresonance case 
$2F(t)\leqslant \mu t^{2}$ with $\mu<\mu_1$ for $|t|\leqslant \delta$ 
as a special case.

\section{Proofs of main results}

 In this section we give the proofs of Theorems \ref{thm1.1} and
\ref{thm1.2}.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
(a) By Lemma \ref{lem3.2}, the functional $J$ satisfies the Palais-Smale 
condition and
\begin{equation}
C_q(\mathcal{J}, \infty)\cong0 \quad \text{for all } q \in \mathbb{Z}.\label{e5.1}
\end{equation}
By Lemma \ref{lem4.2}, we have that
\begin{equation}
C_{\ell_0}(\mathcal{J}, 0) \not \cong 0.\label{e5.2}
\end{equation}
It follows that
\begin{equation}
C_{\ell_0}(\mathcal{J}, \infty) \not \cong C_{\ell_0}(\mathcal{J}, 0).\label{e5.3}
\end{equation}
Therefore $\mathcal{J}$ has at least one nontrivial critical point.

 (c) By Lemma \ref{lem3.3}, the functional $J$ satisfies the Cerami condition and
\begin{equation}
C_\ell(\mathcal{J}, \infty)\not\cong 0 \quad \text{for }
 \ell=\ell_\infty  \text{ or }
 \ell=\ell^*_\infty.\label{e5.4}
\end{equation}
By Lemma \ref{lem4.1}, we have
\begin{equation}
C_q(\mathcal{J}, 0) \cong 0 \quad \text{for all }
 q \in \mathbb{Z}.\label{e5.5}
\end{equation}
It follows that
\begin{equation}
C_{\ell}(\mathcal{J}, \infty) \not \cong C_{\ell}(\mathcal{J}, 0).\label{e5.6}
\end{equation}
Therefore $\mathcal{J}$ has at least one nontrivial critical point.

 The other cases are proved in a similar way.
 The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 We give the proof for the case (b). By Lemma \ref{lem3.4}, $\mathcal{J}$ 
is coercive on $H^1_{0, L}(\Omega)$ and satisfies the Palais-Smale condition.
 Thus $\mathcal{J}$ is bounded from below and has a global minimizer.
 By Lemma \ref{lem4.2}, we have that 
\begin{equation} 
C_{\ell_0}(\mathcal{J}, 0)\not\cong0.\label{e5.7}
\end{equation}
Since $\ell_0 \geqslant 1$, the trivial critical point $0$
 is homological nontrivial and is not a minimizer of $\mathcal{J}$.
It follows from Proposition \ref{prop2.6} that $\mathcal{J}$ has at
least two nontrivial critical points. The proof is complete.
\end{proof}
 
\subsection*{Acknowledgments}
This research was supported by the KZ201510028032 and NSFC
(11771302,11601353,11671026).

The authors want to thank the anonymous referees
 for valuable comments and suggestions.


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