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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 266, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/266\hfil Infinitely many sign-changing solutions]
{Infinitely many sign-changing solutions for concave-convex
elliptic problem with nonlinear boundary condition}

\author[L. Wang, P. Zhao \hfil EJDE-2015/266\hfilneg]
{Li Wang, Peihao Zhao}

\address{Li Wang \newline
School of Mathematics and Statistics,
 Lanzhou University, Lanzhou 730000, China}
\email{lwang10@lzu.edu.cn}

\address{Peihao Zhao\newline
School of Mathematics and Statistics,
 Lanzhou University, Lanzhou 730000, China}
\email{zhaoph@lzu.edu.cn}

\thanks{Submitted August 30, 2015. Published October 16, 2015.}
\subjclass[2010]{35J60, 47J30, 58E05}
\keywords{Nonlinear boundary condition; concave-convex; invariant sets;
\hfill\break\indent  sign-changing solutions}

\begin{abstract}
 In this article, we study the existence of sign-changing
 solutions to
 \begin{gather*}
 -\Delta u+u  =|u|^{p-1}u\quad \text{in } \Omega \\
 \frac{\partial u}{\partial n}=\lambda |u|^{q-1}u\quad \text{on }\partial \Omega
 \end{gather*}
 with $0<q<1<p\leq \frac{N+2}{N-2}$ and $\lambda>0$.
 By using a combination of invariant sets and Ljusternik-Schnirelman
 type minimax method, we obtain two sequences of sign-changing solutions when
 $p$ is subcritical and one sequence of sign-changing solutions when $p$ is critical.
\end{abstract}

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

\section{Introduction}

In this article we study the existence of infinitely many sign-changing 
solutions to the  nonlinear Neumann problem
\begin{equation}\label{q1}
\begin{gathered}
  -\Delta u+u  =|u|^{p-1}u,\quad \text{in } \Omega \\
  \frac{\partial u}{\partial n}=\lambda |u|^{q-1}u,\quad \text{on } \partial \Omega,
  \end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary 
$\partial \Omega$, $N>2$, $\frac{\partial}{\partial n}$ denotes the outward 
normal derivative and $0<q<1<p$, $\lambda>0$. 

The existence of sign-changing solutions has been studied extensively 
in recent years. For the Dirichlet problem
\begin{align}\label{P}
\begin{gathered}
  -\Delta u=f(u) \quad\text{in }\Omega  \\
  u=0  \quad\text{on }\partial\Omega
  \end{gathered}
\end{align}
the authors in \cite{Bartsch:96} considered that for
 $f\in C^{1}(\mathbb{R})$, $f(0)=0$ and $
\lim_{u\to\infty}f'(u)<\lambda_1<\lambda_2<f'(0)$, in which
$\lambda_{i}$ is the eigenvalue of $-\Delta$ on $\Omega$, then  
problem \eqref{P} has at least one sign-changing solution. 
If $f(u)$ is odd about $u$, superlinear and subcrtical, 
Bartsch  \cite{Bartsch:01}
Showed that problem \eqref{P} has a sequence of unbounded sign-changing 
solutions. 
In this case, the positive cone is a invariant set of gradient flow.  
For $f(u)=\lambda u+|u|^{2^{*}-2}u$, $N\geq7$, $\lambda>0$, the authors 
in \cite{Schechter:10,Sun:14} proved that \eqref{P} has 
also infinitely many sign-changing solutions.
 We can look for more examples in 
\cite{Bartsch:96,Bartsch:01,Bartsch:00,Bartsch:99} 
and references therein.
Problems with nonlinear boundary condition of form \eqref{q1} appear in 
a nature way when one considers the Sobolev trace embedding 
$H^{1}(\Omega)\hookrightarrow L^{q}(\partial \Omega)$ and conformal 
deformations on Riemannian manifolds with boundary, 
see \cite{Escobar:90,Escobar:92}. In \cite{Garcia:04}, 
Garcia et al  considered  problem \eqref{q1}. 
For subcritical case, $0<q<1<p<\frac{N+2}{N-2}$, there exists a 
$\lambda_{0}>0$ such that if $0<\lambda<\lambda_{0}$, 
 equation \eqref{q1} has infinitely many solutions with negative energy; 
and for $0<q<1<p\leq\frac{N+2}{N-2}$, there exists $\Lambda>0$, 
such that for $0<\lambda<\Lambda$, there exists at least two 
positive solutions for \eqref{q1}, for $\lambda=\Lambda$, 
at least one positive solution, and no positive solution 
for $\lambda>\Lambda$.
Kajikiya et al \cite{Kajikiya:14} studied the problem
\begin{equation}\label{q2}
  \begin{gathered}
  -\Delta u+u  =f(x,u)\quad \text{in }\Omega, \\
  \frac{\partial u}{\partial n}=g(x,u),\quad \text{on } \partial \Omega.
  \end{gathered}
\end{equation}
and proved that the problem has two sequences of solutions,  
if $f(x,u)$ and $g(x,u)$ satisfying that in a neighborhood of $u=0$, 
one of $f(x,u)$ and $g(x,u)$ is locally sublinear, and at infinity, one 
of them is locally superlinear, and they showed that one sequence of 
the solutions converges to 0, the other diverges to infinity.

Inspired by \cite{Garcia:04,Kajikiya:14}, we consider the existence 
of sign-changing solutions of problem \eqref{q1}. The main part of our
 work is that  \eqref{q1} has two sequence of sign-changing solutions under 
the subcritical and concave case. In this sense, the work of the present 
paper extends the results of \cite{Garcia:04,Kajikiya:14} partially.
In section 2, we give the main results of the paper. 
In section 3,  we establish the invariant sets of pseudo gradient.
In section 4, we proof the theorems.

\section{Main results}

In this section, we state the main results and some preliminaries.
 We call $u$ a weak solution of \eqref{q1} if $u\in H^{1}(\Omega)$
and it satisfies \eqref{q1} in the distribution sense, i.e.
$$
\int_{\Omega}(\nabla u\nabla v+uv)dx=\int_{\Omega}|u|^{p-1}uvdx
+\lambda\int_{\partial\Omega}|u|^{q-1}uvd\sigma,
$$
for any $v\in ~H^{1}(\Omega)$. Here $d\sigma$ denotes the surface measure on 
$\partial \Omega$.

Throughout this paper, the norm of $H^{1}(\Omega)=W^{1,2}(\Omega)$ is defined by
$$
\|u\|:=\Big(\int_{\Omega}(|\nabla u|^{2}+u^{2})dx\Big)^{1/2},
$$
and the $H^{1}(\Omega)$ inner product of $u$ and $v$ by
$$
(u,v):=\int_{\Omega}(\nabla u\nabla v+uv)dx.
$$
We state the  main result as follows.

\begin{theorem}\label{the2.1}
For $0<q<1<p<\frac{N+2}{N-2}$, $\lambda>0$, there exist at least two 
sequences of sign-changing solutions of \eqref{q1}, one  converges 
to 0 in $H^{1}(\Omega)$, and the other diverges to infinity.
\end{theorem}

\begin{theorem}\label{the2.2}
For $0<q<1,~p=\frac{N+2}{N-2}$, $\lambda>0$, there exists at least one sequence 
of sign-changing solutions of \eqref{q1} which converges to 0 in $H^{1}(\Omega)$.
\end{theorem}

Now, we state some results we will need in the following sections. 
Next lemma is Lemma 6.1 in \cite{Garcia:04}.

\begin{lemma}\label{l2.1}
For $0<q<1<p\leq \frac{N+2}{N-2}$, there exists a $\Lambda>0$ such that for
 $\lambda\leq\Lambda$, then \eqref{q1} has a minimal positive solution $u^{+}$ 
and a maximal negative solution $u_{-}$.
\end{lemma}

If $\lambda>\Lambda$, \eqref{q1} has no positive and negative solutions, 
by the results in \cite{Kajikiya:14} as we mentioned above,
we know the existence of two sequences of solutions that are sign-changing solutions. 
Hence, we only need to prove the results in Theorem \ref{the2.1} under the 
condition $0<\lambda\leq\Lambda$. Throughout the paper, we assume that 
$\lambda\leq\Lambda$.
For \eqref{q1} the minimal positive solution and the maximal negative 
solution satisfying $u^{+}=-u^{-}$.

The following lemma is a variant of \cite[Lemma 3.2]{Liu:2001} 
and we can also look for \cite[Lemma 2.4]{Liu:2005}.

\begin{lemma}\label{l2.2}
Let $H$ be a Hilbert space, $D_1$ and $D_2$ be two closed convex subsets of
$H$, and $I\in C^{1}(H,\mathbb{R})$. Suppose $I'(u)=u-A(u)$ and 
$A(D_{i})\subset D_{i}$ for $i=1,2$. Then there exists a pseudo gradient
 vector field $V$ of $I$ in the form $V(u)=u-B(u)$ with $B$ satisfying 
$B(D_{i})\subset \operatorname{int}(D_{i})$ if 
$A(D_{i})\subset \operatorname{int}(D_{i})$ for $i=1,2$, and 
$V$ is odd if $I$ is even and $D_1=-D_2$.
\end{lemma}

We refer to \cite{Kajikiya:14} for the following priori estimates.

\begin{lemma}\label{l2.3}
Let $f(x,s)$ and $g(x,s)$ satisfy:
\begin{itemize}
\item[(1)] $|f(x,s)|\leq C(|s|^{p}+1)$,
\item[(2)] $|g(x,s)|\leq C(|s|^{q}+1)$ with $0<q<1<p<\frac{N+2}{N-2}$
\end{itemize}
Then for every $H^{1}(\Omega)$-solution $u$ of \eqref{q2} belongs 
to $W^{1,r}(\Omega)$ for all $r<\infty$, and satisfies
$$
\|u\|_{W^{1,r}(\Omega)}\leq C_{r}\|u\|^{dp}_{H^{1}(\Omega)}
+C_{r}\|u\|^{dq}_{H^{1}(\Omega)}+C_{r},
$$
where $C_{r}$ is a constant depends only on $r$ and $d$ is independent 
of $u$ and $r$.
\end{lemma}

Here we call that $V$ is a pseudo gradient vector field of $I$ if $V\in C(H,H)$, 
$V|_{H\backslash K}$is locally Lipschitz continuous with $K:=\{u\in H:I'(u)=0\}$, 
and $(I'(u),V(u))\geq\frac{1}{2}\|I'(u)\|^{2}$ and 
$\|V(u)\|\leq 2\|I'(u)\|$ for all $u\in H$.

\section{Invariant sets of the gradient flow}

To construct nodal solutions we need to isolate the signed solutions into 
certain invariant sets. We know that  problem \eqref{q1} has a minimal 
positive solution and a maximal negative solution, by this results
 we can build the invariant sets.

Define $v:=A(u)$, $u\in H^{1}(\Omega)$ if
\begin{gather*}
  -\Delta v+v  =|u|^{p-1}u\quad \text{in } \Omega \\
  \frac{\partial v}{\partial n}=\lambda |u|^{q-1}u\quad \text{on } \partial \Omega,
\end{gather*}
and
\begin{gather*}
 \frac{d}{dt}\eta^{t}(u)= -\eta^{t}(u)+B(\eta^{t}(u)) \\
  \eta^{0}(u)=u ,
\end{gather*}
where $B$ is related to $A$ via Lemma \ref{l2.2} in which $D_1$ 
and $D_2$ will be constructed in Theorem \ref{the3.1}. 
This section is concerned with the construction of these sets which are
 invariant under the flow $\eta^{t}(u)$ such that all positive and negative 
solutions are contained in these invariant sets. 
Recall that a subset $W\subset H$ is an invariant set with respect to $\eta$ if, 
for any $u\in W$, $\eta^{t}(u)\in W$ for all $t>0$.

We first note that because of the sublinear term on the boundary, any 
neighborhoods of the positive (and negative) cones are no longer invariant 
sets of the gradient flow. We give a construction inspired by \cite{Liu:2005}.
Let $e_1\in H^{1}(\Omega)$ be the first eigenfunction associated with the 
first eigenvalue $\lambda_1$ of the eigenvalue problem
\begin{equation} \label{3.1}
\begin{gathered}
      -\Delta u+u =0 \quad \text{in }\Omega  \\
     \frac{\partial u}{\partial n}=\lambda u \quad \text{on }
\partial \Omega 
\end{gathered}
\end{equation}
such that $\max_{\Omega} e_1(x)\leq s_{0}$, in which 
$0<s_{0}\leq\min_{\Omega}u^{+}(x)$ and $s_{0}$ to be determined later.
Then we have
$u^{+}(x)\geq e_1(x)$ and  $u^{-}(x)\leq-e_1(x) $
for all $x\in \Omega$, $u^{\pm}$ is the minimal positive solution and 
maximal negative solution to \eqref{q1}, respectively. Define:
$$
D^{\pm}:=\{u\in H^{1}(\Omega):\pm u\geq e_1\}.
$$
From above, we know that all positive and negative solutions to \eqref{q1} 
are contained in $D^{+}$ and $D^{-}$, respectively.
Define $(D^{\pm})_{\epsilon}=\{u\in H^{1}(\Omega): 
\operatorname{dist}(u,D^{\pm})<\epsilon\}$.

\begin{theorem}\label{the3.1}
Assume $0<q<1<p<\frac{N+2}{N-2}$, and $0<\lambda\leq\Lambda$. 
Then there exists $\epsilon_{0}>0$ such that
\begin{gather*}
A((D^{\pm})_{\epsilon})\subset \operatorname{int}((D^{\pm})_{\epsilon})\quad
\text{for all }0<\epsilon<\epsilon_{0}, \\
\eta^{t}((D^{\pm})_{\epsilon})\subset \operatorname{int}((D^{\pm})_{\epsilon})\quad
\text{for all }t\geq 0,\; 0<\epsilon<\epsilon_{0},
\end{gather*}
\end{theorem}

\begin{proof}
We only prove the result for the positive one, the other case follows analogously.
For $u\in H^{1}(\Omega)$, we denote
\[
v=Au, \quad v_1=\max\{e_1,v\}.
\]
Then $\operatorname{dist}(v,D^{+})\leq \|v-v_1\|$ which implies
$\operatorname{dist}(v,D^{+})\cdot\|v-v_1\|\leq \|v-v_1\|^{2}$ and
\begin{align*}
  \|v-v_1\|^{2} 
&= (v-e_1,v-v_1) \\
  &=\int_{\Omega}\nabla (v-e_1)\cdot\nabla(v-v_1)+(v-e_1)(v-v_1)dx\\
  &=\int_{\Omega}(-\Delta (v-e_1)+v-e_1)(v-v_1)dx
+\int_{\partial\Omega}(\lambda|u|^{q-1}u-\lambda_1e_1) (v-v_1)d\sigma\\
  &=\int_{\Omega}(|u|^{p-1}u)(v-v_1)dx
+\int_{\partial\Omega}(\lambda|u|^{q-1}u-\lambda_1e_1) (v-v_1)d\sigma\\
  &=: I_1+I_2.
\end{align*}
Note that
\begin{align*}
  I_1&\leq\int_{\{u<0\}\cap\Omega}(v_1-v)(-|u|^{p-1}u)dx
\leq  \int_{\{u<0\}\cap\Omega}(v_1-v)(e_1-|u|^{p-1}u)dx\\
   & \leq C_{p}\int_{\{u<0\}\cap\Omega}(v_1-v)(e_1-u)^{p}dx.
\end{align*}
On $\{u<0\}\cap\Omega$, we have $u\leq e_1$, hence
\begin{align*}
\|e_1-u\|^{p}_{L^{p+1}((u<0)\cap\Omega)}
& =\inf_{w\in D^{+}}\|w-u\|^{p}_{L^{p+1}((u<0)\cap\Omega)} \\
&\leq \inf_{w\in D^{+}}\|w-u\|^{p}_{L^{p+1}(\Omega)}\leq C_{p}dist^{p}(u,D^{+}),
\end{align*}
and $I_1\leq C\|v-v_1\|\operatorname{dist}^{p} (u,D^{+})$.
Here $C_{p}$ and $C$ are constants which are relevant to $p$ and $e_1$, 
and may change from line to line.
Note that
\begin{align*}
  I_2 
& \leq \int_{\partial\Omega\cap\{\lambda|u|^{q-1}u<\lambda_1e_1\}}
  (\lambda_1e_1-\lambda|u|^{q-1}u)(v_1-v)d\sigma\\
&=\Big(\int_{\partial\Omega\cap\{\lambda(\frac{e_1}{2})^{q}<\lambda|u|^{q-1}
u<\lambda_1e_1\}}+ \int_{\partial\Omega\cap\{u\leq\frac{e_1}{2}\}}\Big)
(\lambda_1e_1-\lambda|u|^{q-1}u)(v_1-v)d\sigma.
\end{align*}
If $\lambda(e_1/2)^{q}>\lambda_1e_1$, the first term above vanishing, 
this can be done by choose $s_{0}$ small enough such that 
$s_{0}^{1-q}\leq\frac{\lambda}{\lambda_12^{q}}$.
On $\partial\Omega\cap\{u\leq\frac{e_1}{2}\}$, we have
\[
\lambda_1e_1-\lambda|u|^{q-1}u\leq C_{r}(e_1-u)^{r},
\]
where $r\in(1,\frac{N}{N-2})$. 
\begin{align*}
\|e_1-u\|^{r}_{L^{r+1}(\{u<\frac{e_1}{2}\}\cap\partial\Omega)}
& =\inf_{w\in D^{+}} \|w-u\|^{r}_{L^{r+1}(\{u<\frac{e_1}{2}\}\cap\partial\Omega)} \\
&\leq \inf_{w\in D^{+}}\|w-u\|^{r}_{L^{r+1}(\partial\Omega)}
\leq C_{r} \operatorname{dist} ^{r}(u,D^{+}).
\end{align*}
Hence,
$I_2\leq C\|v-v_1\| \operatorname{dist}^{r} (u,D^{+})$.
\[
\operatorname{dist}(v,D^{+})\cdot\|v-v_1\|
\leq C\|v-v_1\|(\operatorname{dist}^{r} (u,D^{+})+ \operatorname{dist}^{p} (u,D^{+}))
\]
Then we can choose $\epsilon_{0}$ small, such that for $\epsilon<\epsilon_{0}$,
\[
\operatorname{dist}(v,D^{+})<\operatorname{dist}(u,D^{+})\quad\text{for }
u\in D^{+}_{\epsilon}.
\]
The first conclusion in Theorem \ref{the3.1} is proved, the second part 
is a consequence of the first one as shown in \cite{Liu:2001} 
via Lemma \ref{l2.2} above.
\end{proof}

\section{Proof of main results}

Let us start with a more abstract setting. 
Consider $I\in C^{1}(X,\mathbb{R})$ where $X$ is a Banach space.
 $V$ is a pseudo gradient vector field of $I$ such that $V$ is odd if $I$ 
is even, and consider
\begin{gather*}
\frac{d}{dt}\sigma(t,u) =-V(\sigma), \\
  \sigma(0,u)=u\in X.
\end{gather*}
To construct nodal solution by using the combination of invariant sets 
and minimax method, we need a deformation lemma in the presence of invariant sets. 
We have the following deformation lemma which follows from 
\cite[Lemma 5.1]{Liu:2005} (see also \cite[Lemma 2.4]{Li:02}).

\begin{lemma}\label{l4.1}
Assume $I$ satisfies the $(PS)$-condition, and $c\in \mathbb{R}$ is fixed, 
$W=\partial W\cup \operatorname{int}(W)$ is an invariant subset 
such that $\sigma(t,\partial W)\subset$int$(W)$ for $t>0$. Define
$K_{c}^{1}:=K_{c}\cap W$, $K_{c}^{2}:=K_{c}\cap(X\backslash W)$, 
where $K_{c}:=\{u\in X:~I'(u)=0,I(u)=c\}$.
Let $\delta>0$, be such that $(K_{c}^{1})_{\delta}\subset W$ where 
$(K_{c}^{1})_{\delta}=\{u\in X:\operatorname{dist}(u,K_{c}^{1})<\delta\}$. 
Then there exists an $\varepsilon_{0}>0$ such that for any 
$0<\varepsilon<\varepsilon_{0}$, there exists 
$\eta\in C([0,1]\times X,X)$ satisfying:
\begin{itemize}
\item[(1)] $\eta(t,u)=u$ for $t=0$ or 
 $u\notin I^{-1}(c-\varepsilon_{0},c+\varepsilon_{0})\backslash(K_{c}^{2})_{\delta}$.

\item[(2)] $\eta(1,I^{c+\varepsilon}\cup W\backslash 
(K_{c}^{2})_{3\delta})\subset I^{c-\varepsilon}\cup W$ and
$\eta(1,I^{c+\varepsilon}\cup W)\subset I^{c-\varepsilon}\cup W$ 
if $K_{c}^{2}=\emptyset$.

\item[(3)] $\eta(t,\cdot)$ is a homeomorphism of X for $t\in [0,1]$.

\item[(4)] $\|\eta(t,u)-u\|\leq \delta$, for any $(t,u)\in [0,1]\times X$.

\item[(5)] $I(\eta(t,\cdot))$ is non-increasing.

\item[(6)] $\eta(t,W)\subset~W$ for any $t\in[0,1]$.

\item[(7)] $\eta(t,\cdot)$ is odd if I is even and if W is symmetric with 
respect to $0$.
\end{itemize}
\end{lemma}

Set
\begin{gather*}
\Sigma:=\{A\subset H^{1}(\Omega)\backslash 0: A \text{ is closed and }A=-A\}, \\
\Gamma_{k}:=\{A\subset H^{1}(\Omega)\backslash 0: A 
\text{is closed, symmetric},\, \gamma(A)\geq k\}
\end{gather*}
where $\gamma(A)$ denotes the Krasnoselskii's genus of the set $A$. 
We refer to \cite{Struwe:90} for the following properties of genus.

\begin{lemma}
Let $A,B \in \Gamma_{k}$, and $h\in C(H^{1}(\Omega),H^{1}(\Omega))$ be an odd map. 
Then
\begin{itemize}
\item[(1)] $A\subset B\Rightarrow~\gamma(A)\leq\gamma(B)$;

\item[(2)] $\gamma(A\cup B)\leq \gamma(A)+\gamma(B)$;

\item[(3)] $\gamma(A)\leq \gamma(h(A))$;

\item[(4)] If $A$ is compact, there exists an $N\in \Gamma_{k}$ such 
that $A\subset \operatorname{int}(N)\subset N$ and $\gamma(A)=\gamma(N)$;

\item[(5)] If $F$ is a linear subspace of $H^{1}(\Omega)$ with $\dim F=$n, 
$A\subset F$ is bounded, open and symmetric, and $0\in A$, then 
$\gamma(\partial_{F}A)=n$;

\item[(6)] Let $W$ be a closed linear subspace of $H^{1}(\Omega)$
 whose codimension is finite. If $\gamma(A)$ is greater than the codimension 
of $W$, then $A\cap W\neq \emptyset$.
\end{itemize}
\end{lemma}

We choose an even function $h\in C^{\infty}_{0}(\mathbb{R})$ such that 
$h(s)=1$ for $|s|\leq 1$, $h(s)=0$ for$|s|\geq 2$, $0\leq h\leq 1$; defining
\begin{gather}\label{tran4.1}
  f(s):=s|s|^{p-1}h(s),~~g(s)=s|s|^{q-1}h(s); \\
\widetilde{I}(u)=\frac{1}{2}\|u\|^{2}-\int_{\Omega}F(u)dx
 -\int_{\partial \Omega}G(u)d\sigma,\nonumber
\end{gather}
in which $F(u)=\int_{0}^{u}f(s)ds$,
 $G(u)=\int_{0}^{u}g(s)ds$, both of them are bounded.
Assume $(\lambda_{i},e_{i})$ is the eigenvalue and corresponding 
eigenfunction of \eqref{3.1}, and $E_{m}=\operatorname{span}\{e_1,\cdots,e_{m}\}$. 
Then the following lemma is obvious.

\begin{lemma}\label{l4.2}
$\widetilde{I}\in C^{1}(H^{1}(\Omega),\mathbb{R})$, 
\begin{itemize}
\item[(1)] for all $m\in \mathbb{N}$, there exists a $\rho>0$, such that 
$\sup_{E_{m}\cap\partial B_{\rho}}\widetilde{I}(u)<0$, where 
$\partial B_{\rho}:=\{u\in H^{1}(\Omega):\|u\|=\rho\}$,

\item[(2)] $\widetilde{I}$ is even, bounded from blow, and the (PS)-condition 
holds, $\widetilde{I}(0)=0$;
\end{itemize}
\end{lemma}

The following lemma is similar to \cite[Lemma 5.3]{Liu:2005}.

\begin{lemma}\label{l4.3}
For any $\rho>0$, let $B_{\rho}=\{u\in H^{1}(\Omega),\|u\|\leq \rho\}$. Then 
\[
  \operatorname{dist}(\partial B_{\rho}\cap E_1^{\bot}, D^{+}\cup D^{-})>0.
\]
\end{lemma}

\begin{proof}
Assume on the contrary, that there exists $(u_{n})\in D^{+}$, 
$v_{n}\in \partial B_{\rho}\cap E_1^{\bot}$, such that
 $\|u_{n}-v_{n}\|\to 0$. Then $(u_{n},e_1)=(u_{n}-v_{n},e_1)+(v_{n},e_1)\to 0$, 
as $n\to \infty$.
But, since $u_{n}\geq e_1$, we have
$$
(u_{n},e_1)=\lambda_1\int_{\partial\Omega}u_{n}e_1
\geq\lambda_1\int_{\partial\Omega}e_1^{2}d\sigma\neq 0,
$$
a contradiction.
\end{proof}

\subsection*{Proof of Theorems}
 We essentially follow from \cite{Liu:2005}, see also \cite{Bartsch:05} 
and \cite{Rabinowitz:86}.
\smallskip

\noindent\textbf{Part 1.} 
In this part, we will prove that for $0<q<1<p\leq\frac{N+2}{N-2} $, 
\eqref{q1} has a sequence of sign-changing solutions which converge 
to $0$. This is a conclusion of \cite{Kajikiya:05} and \cite{Garcia:04}.
By Lemma \ref{l4.2} above we have taht
for each $k\in \mathbb{N}$, there exists an
 $A_{k}\in \Gamma_{k}$ such that $\sup_{u\in A_{k}}\widetilde{I}(u)<0$.
With the help of \cite[Theorem 1]{Kajikiya:05}, there exists a sequence 
$\{u_{k}\}$ satisfying
\[
\widetilde{I'}(u_{k})=0,\quad \widetilde{I}(u_{k})<0, \quad
u_{k} \to 0 \text{ in }H^{1}(\Omega).
\]
By Lemma \ref{l2.3}, $u_{k}$ converges to zero in $C(\overline{\Omega})$. 
Hence for large $k$, we have $\|u_{k}\|_{C(\overline{\Omega})}<1$, 
$\widetilde{I}(u_{k})=I(u_{k})$ and $\widetilde{I'}(u_{k})=I'(u_{k})$.
But from Lemma \ref{l2.1} we know that \eqref{q1} has a minimal positive 
solution and a maximal negative solution, thus, for large $j$, $u_{j}$ 
must change signs.
Theorem \ref{the2.2} and the first part of Theorem \ref{the2.1} 
follows from the above argument.
\smallskip

\noindent\textbf{Part 2.}
 In this part, we  prove the existence of a sequence of sign-changing 
solutions which tends to infinity under the case $0<q<1<p<\frac{N+2}{N-2}$.
The functional
$$
I(u)=\frac{1}{2}\|u\|^{2}-\frac{1}{p+1}\int_{\Omega}|u|^{p+1}dx
-\frac{\lambda}{q+1}\int_{\partial\Omega}|u|^{q+1}d\sigma,
$$
is well defined on $H^{1}(\Omega)$ and $I\in C^{1}(H^{1}(\Omega),\mathbb{R})$, 
$I$ satisfies the (PS) condition for $0<q<1<p<\frac{N+2}{N-2}$.

\begin{lemma}\label{l4.4}
Assume $m\geq 2$, then there exists $R=R(m)>0$ such that for all $\lambda>0$,
$$
\sup_{B_{R}^{c}\cap E_{m}}I(u)<0.
$$
where $B_{R}^{c}:=H^{1}(\Omega)\backslash B_{R}$.
\end{lemma}

From Theorem \ref{the3.1} we can choose an $\epsilon>0$ small enough such that 
$(D^{\pm})_{\epsilon}$ are invariant sets. 
Set W=$\overline{(D^{+})_{\epsilon}}\cup\overline{(D^{-})_{\epsilon}}$, 
$S=H^{1}(\Omega)\backslash W$ contains only sign-changing solutions. 
Set
\[
G_{m}=\{h\in C(B_{R}\cap E_{m},H^{1}(\Omega)):h\text{ is odd and $h=$id
on }\partial B_{R}\cap E_{m} \},
\]
in which $R$ is determined in Lemma \ref{l4.4}.
\[
\widetilde{\Gamma}_{j}=\{h(\overline{B_{R}\cap E_{m}\backslash Y}):
h\in G_{m}, \,\forall m\geq j,\, Y=-Y, \text{ closed, }
\gamma(Y)\leq m-j\},\quad j\geq 2.
\]
From \cite{Ambrosetti:73} and\cite{Liu:2005}, we know that 
$\widetilde{\Gamma}_{j}$ satisfying the following properties:
\begin{itemize}
\item[(1')]  $\widetilde{\Gamma}_{j}\neq \emptyset$ for all $j\geq 2$.

\item[(2')] $\widetilde{\Gamma}_{j+1}\subset \widetilde{\Gamma}_{j}$ for all 
$j\geq 2$.

\item[(3')] if $\sigma\in C(H^{1}(\Omega),H^{1}(\Omega))$ is odd and 
$\sigma=id$ on $\partial B_{R}\cap E_{m}$, then 
$\sigma(A)\in \widetilde{\Gamma}_{j}$ if $A\in \widetilde{\Gamma}_{j}$. 

\item[(4')] if $A\in \widetilde{\Gamma}_{j}$, $Z=-Z$, closed, and 
$\gamma(Z)\leq s<j$ and $j-s\geq 2$, then 
$\overline{A\backslash Z}\in \widetilde{\Gamma}_{j-s}$.
\end{itemize}
For $j\geq 2$, we define
$$
\widetilde{c}_{j}:=\inf_{A\in \widetilde{\Gamma}_{j}}\sup_{u\in A\cap S}I(u).
$$
If $A\in \widetilde{\Gamma}_{j}$ with $j\geq 2$, then 
$A\cap\partial B_{\rho}\cap(E_1)^{\bot}\neq \emptyset$.
By Lemma \ref{l4.3}, $\partial B_{\rho}\cap(E_1)^{\bot}\subset S$. 
Thus, for $j\geq 2$, and $A\in\widetilde{\Gamma}_{j}$, $A\cap S\neq \emptyset$, 
we conclude that
$$
\widetilde{c}_{j}\geq \inf_{\partial B_{\rho}\cap(E_1)^{\bot}}I(u)>-\infty.
$$
Then from the definition of $\widetilde{c}_{j}$ and (2') we have 
$-\infty<\widetilde{c}_2\leq \widetilde{c}_{3}\leq
\cdots\leq\widetilde{c}_{j}\leq\cdots<\infty$.
 We claim that if $c:=\widetilde{c}_{j}=\cdots=\widetilde{c}_{j+k}$ 
for some  $2\leq j\leq j+k$ with $k\geq 0$, then 
$\gamma(K_{c}\cap S)\geq k+1$. 
Before we prove this claim, we first show that
 $\widetilde{c}_{j}\to \infty$, as $j\to\infty$.
We need the following lemma.

\begin{lemma}
The constant $\widetilde{c}_{j}$ is independent of the choice of $R(m)$ 
as long as $R(m)$  is chosen to satisfy
Lemma \ref{l4.4} for which $m\geq j$.
\end{lemma}

The above lemma is well known, see for instance \cite[Lemma 4.9]{Kajikiya:10}.
And we can choose $R(m)$ such that $R(m)\to \infty$, as $m\to \infty$. 
This part follows by \cite{Kajikiya:14}.
Let $W_{m}:=\{\sum_{i=m}^{\infty}t_{i}w_{i}:\sum_{i=m}^{\infty}t_{i}^{2}<\infty\}$ 
and $w_{m}$ is the eigenfunction of the Neumann Laplacian equation:
\[
-\Delta w= \mu w \quad \text{in } \Omega,\quad 
\frac{\partial w}{\partial n}=0\quad\text{on }\partial \Omega.
\]
$W_{j}$ is a closed linear subspace of $H^{1}(\Omega)$ whose codimension is equal 
to $j-1$, we have:
$$
h(\overline{B_{R}\cap E_{m}\backslash Y})\cap \partial B_{r}\cap W_{j}
\neq \emptyset,
$$
for $h\in G_{m}$, $\gamma(Y)\leq m-j$, and $0<r<R$(since that 
$\gamma(h(\overline{B_{R}\cap E_{m}\backslash Y}))\geq j$, and the codimension 
of $W_{j}$ is $j-1$. This implies 
$$
\sup_{u\in B_{R}\cap E_{m}\backslash Y}I(h(u))\geq \inf\{I(u):
u\in \partial B_{r}\cap W_{j}\},
$$
for $h\in G_{m}$, $\gamma(Y)\leq m-j$, taking the infimum of both sides over 
$h\in G_{m}$, we have
$$
\widetilde{c}_{j}\geq inf\{I(u):~u\in \partial B_{r}\cap W_{j}\},
$$
for $0<r<R$. Next we can have $\inf\{I(u):u\in \partial B_{r}\cap W_{j}\}$ 
diverges to $\infty$, the rest of the proof is similarly with 
\cite[Lemma 5.14]{Kajikiya:14}, we omit it here.

Now we give the proof the claim. Denote $K_{c}\cap S$ by 
$K_{c}^{2}$. If the claim is false, $\gamma(K_{c}\cap S)\leq k$, because of 
$\widetilde{c_{j}}\to \infty$, we can assume that $0\notin K_{c}$ and 
$K_{c}^{2}=K_{c}\cap S$ is compact, there exists $N$ such that 
$K_{c}^{2}\subset \operatorname{int}(N)$ and $\gamma(N)=\gamma(K_{c}^{2})$.
Then by Lemma \ref{l4.1}, there exists an $\epsilon_{0}>0$ such that for 
$0<\epsilon<\epsilon_{0}$, there exists an 
$\eta\in C([0,1]\times H^{1}(\Omega),H^{1}(\Omega))$ satisfying (1)-(7)
 of Lemma \ref{l4.1}. Then
$$
\eta(1,I^{c+\epsilon}\cup W\backslash N)\subset (I^{c-\epsilon}\cup W).
$$
Choose $A\in \widetilde{\Gamma}_{j+k}$ such that
$$
\sup_{A\cap S}I(u)\leq c+\epsilon,
$$
Then by (4') above $\overline{A\backslash N}\in \widetilde{\Gamma}_{j}$
hence $\eta(1,\overline{A\backslash N})\in \widetilde{\Gamma}_{j}$. Then
$$
c\leq\sup_{\eta(1,\overline{A\backslash N})}I(u)\leq\sup_{(I^{c-\epsilon}\cup W)
\cap S}\leq c-\epsilon,
$$
contradiction. Hence $\gamma(K_{c}\cap S)\geq k+1$. Now we finish the proof.

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