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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 185, pp. 1--9.\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-2017/185\hfil Existence of solutions]
{Existence of solutions to $(2,p)$-Laplacian equations by Morse theory}

\author[Z. Liang, Y. Song, J. Su \hfil EJDE-2017/185\hfilneg]
{Zhanping Liang, Yuanmin Song, Jiabao Su}

\address{Zhanping Liang \newline
School of Mathematical Sciences,
Shanxi University, Taiyuan 030006, Shanxi, China}
\email{lzp@sxu.edu.cn}

\address{Yuanmin Song \newline
School of Mathematical Sciences,
Shanxi University, Taiyuan 030006, Shanxi, China}
\email{954126769@qq.com}

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

\dedicatory{Communicated by Marco Squassina}

\thanks{Submitted September 26, 2016. Published July 21, 2017.}
\subjclass[2010]{35J50, 35J92, 58E05}
\keywords{$(2,p)$-Laplacian equation; Morse theory}

\begin{abstract}
 In this article, we use Morse theory to investigate a type of Dirichlet
 boundary value problem related to the $(2,p)$-Laplacian operator,
 where the nonlinear term  is characterized by the first eigenvalue of
 the Laplace operator. The investigation is heavily  based on a new
 decomposition about the Banach space $W^{1,p}_0(\Omega)$, where
 $\Omega\subset \mathbb{R}^N$ $(N\geqslant 1)$  is a bounded domain with smooth
 enough boundary.

\end{abstract}

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

\section{Introduction and statement of main results}

Recently, much attention  has been paid to the existence of solutions to the
quasilinear elliptic problems of $(q,p)$-Laplacian type
\begin{equation}\label{1.50}
\begin{gathered}
-\Delta_qu-\Delta_pu=h(x,u),\quad x\in\Omega,\\
u=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where  $\Omega\subset \mathbb{R}^N(N\geqslant 1)$ is a bounded domain with smooth
boundary $\partial \Omega$,
$\Delta_q u=\operatorname{div}(|\nabla u|^{q-2}\nabla u)$ and 
$\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ are respectively  
the $q$-Laplacian and $p$-Laplacian of $u$. Solutions to \eqref{1.50} are 
the steady state solutions of  the general reaction-diffusion equation
\begin{equation}\label{1.2}
u_t=\operatorname{div}\left(H(u)\nabla u\right)+h(x,u),
\end{equation}
where $H(u)=|\nabla u|^{q-2}+|\nabla u|^{p-2}$. Equation \eqref{1.2} 
has a wide range of applications in physics and related sciences such as 
biophysics \cite{fife}, plasma physics \cite{struwe} and chemical reaction 
design \cite{aris}. The stationary solutions to \eqref{1.2} have been 
studied by many authors using variational methods; see 
\cite{Candito,han, Marano,sidiropoulos,Yin,zhang}.

In this article, we use  Morse theory  to show the existence of solutions
 to the  (2,$p$)-Laplacian equation
\begin{equation}\label{1.1}
\begin{gathered}
-\Delta u-\Delta_p u=f(x,u),\quad x\in\Omega,\\
u=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where  $p>2$. This work is motivated by our previous research of \eqref{1.1}; 
see \cite{han,zhang}. Assume that the nonlinear term $f$ satisfies the following 
hypotheses.
\begin{itemize}
\item[(H1)] $f\in C(\overline{\Omega}\times \mathbb{R},\mathbb{R}),\ f(x,t)\geqslant0$ 
for all $x\in\overline{\Omega},\ t>0$ and $f(x,t)=0$ for all 
 $x\in\overline{\Omega},\ t\leqslant0$.

\item[(H2)] For $f_0,\ f_\infty<\infty$, the limits
\begin{equation*}
\lim_{t\to 0^+}\frac{f(x,t)}t=f_0,\quad
 \lim_{t\to\infty}\frac{f(x,t)}{t^{p-1}}=f_\infty
\end{equation*}
exist uniformly for $x\in\overline{\Omega}$.
\end{itemize}
In \cite{zhang}, we show that if $f$ satisfies (H1) and (H2) 
with $f_0<\lambda_1$ and $\ f_\infty>\mu_1$, then \eqref{1.1} has a positive 
solution, where
\begin{gather*}
\lambda_1=\inf\Big\{\int_\Omega|\nabla u|^2:u\in H_0^1(\Omega),
\int_\Omega|u|^2=1\Big\}, \\
\mu_1=\inf\Big\{\int_\Omega|\nabla u|^p:u\in W^{1,p}_0(\Omega),
\int_\Omega|u|^p=1\Big\}.
\end{gather*}
 What about the situation that $f_0>\lambda_1$ or $f_\infty<\mu_1$? 
In \cite{han}, we have studied the case $f_\infty<\mu_1$ and obtain the 
existence result of non-negative solutions to \eqref{1.1}. 
But we can not guarantee the existence of nontrivial solutions to \eqref{1.1}; 
see \cite[Proposition 1.3]{han}. In this article, we pour our attention 
into the situation that $f_0>\lambda_1$ and give some spirit for the existence
 of nontrivial solutions to \eqref{1.1}. Now, we give some assumptions 
for the nonlinearity $f$ of the present article.
\begin{itemize}
\item[(H3)]  $f\in C(\overline{\Omega}\times \mathbb{R}, \mathbb{R})$ and there
exist a constant $C>0$ and $q\in (p,p^*)$ such that for all
$x\in\overline{\Omega}$, $t\in\mathbb{R}$,
\begin{equation*}
|f(x,t)|\leqslant C(1+|t|^{q-1}),
\end{equation*}
  where $p^*=Np/(N-p)$ if $N>p$ and $p^*=\infty$ if $N\leqslant p$.

\item[(H4)] The following limit holds uniformly for $x\in \overline{\Omega}$,
\begin{equation*}
\lim_{|t|\to \infty} \frac{F(x,t)}{|t|^p}=+\infty,
\end{equation*}
  where $F(x,t)=\int^t_0 f(x,s){\rm d}s$.

\item[(H5)] There exists $R>0$ such that for all $x\in \overline{\Omega}$, 
$\frac{f(x,t)}{|t|^{p-2}t}$ is increasing for $t\geqslant R$ and decreasing 
for $t\leqslant -R$.

\item[(H6)] There exists $l>0$ such that
\begin{equation*}
\lim_{t\to 0} \frac{f(x,t)}{t}=l
\end{equation*}
 uniformly for $x\in\overline{\Omega}$.
\end{itemize}

Our main result is the following theorem.

\begin{theorem}\label{thm1.1}
Suppose that {\rm (H3)--(H6)} are satisfied with $l>\lambda_1$ and 
$l\not\in\sigma(\Delta)$, where $\sigma(\Delta)$ is the spectral set of 
$(-\Delta,H^1_0(\Omega))$. Then  \eqref{1.1} has at least one nontrivial solution.
\end{theorem}

Let $X$ be a real Banach space and $I\in C^1(X, \mathbb{R})$.
We say that $\{u_n\}\subset X$ is a Cerami sequence if $\{I(u_n)\}$ 
is bounded and $(1+\|u_n\|)\|I'(u_n)\|\to 0$ as $n\to\infty$ and $I$ 
satisfies the Cerami condition if any Cerami sequence has a convergent subsequence.
Denote by $C_k(I, u)$ and $C_k(I,\infty)$  the kth critical group of $I$ at 
an isolated critical point $u$ and the kth   critical group of $I$ at 
infinity respectively.

To prove Theorem \ref{thm1.1}, we use the following theorem.

\begin{theorem}[\cite{g,h}] \label{thm1.2}
Suppose that $I\in C^1(X,\mathbb{R})$ satisfies the Cerami condition and $I$ has 
only finitely many critical points. Let $\theta$, the zero element of $X$, 
be a critical point of $I$. If for some $k\in \mathbb{N}$ we have 
$C_k(I, \theta)\neq C_k(I,\infty)$, then $I$ has a nonzero critical point.
\end{theorem}

\section{Proof of main results}

First, we introduce some notation. Let
\begin{equation*}
X=W^{1,p}_0(\Omega),\quad
\|u\|=\Big(\int_\Omega|\nabla u|^p\Big)^{1/p},\quad
 \|u\|_H=\Big(\int_{\Omega}|\nabla u|^2\Big)^{1/2}.
\end{equation*}
$X^*$ is the dual space of $X$ and $c_1,c_2,\dots$ denote various positive 
constants whose exact values are not essential to the analysis of the 
relevant problems. $|D|$ means the Lebesgue measure of Lebesgue measurable set $D$.  
Define the energy functional $I: X\to \mathbb{R}$ by
\begin{equation*}
I(u)=\frac{1}{2}\int_\Omega |\nabla u|^2+\frac{1}{p}\int_\Omega |\nabla u|^p
-\int_\Omega F(x, u),\quad u\in X.
\end{equation*}
It is obvious that the functional $I$ is well defined and belongs to $C^1(X, \mathbb{R})$;
see \cite{han,R}. Furthermore,
\begin{equation*}
\langle I'(u),v\rangle=\int_\Omega \nabla u\cdot \nabla v
+\int_\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla v
-\int_\Omega f(x,u)v,\quad u,v\in X,
\end{equation*}
where $\langle\cdot, \cdot\rangle$ denotes the duality pairing between $X^*$ 
and $X$. Clearly, critical points of $I$ are the weak solutions to  \eqref{1.1}.

\begin{lemma}\label{lem2.1}
Assume that {\rm (H3)--(H6)} are satisfied. Then $I$ satisfies the Cerami condition.
\end{lemma}

\begin{proof}
By (H3), (H4) and (H6), we know that there exists $\mu>0$ such that for all 
$(x,t)\in \overline{\Omega}\times \mathbb{R}$,
\begin{equation}\label{2.1}
F(x,t)\geqslant -\mu|t|^p.
\end{equation}
Letting $\{u_n\}$ be a Cerami sequence of $I$, we first prove that $\{u_n\}$ 
is bounded in $X$. If $\{u_n\}$ is unbounded, up to a subsequence, then we 
may assume that for some $c\in \mathbb{R}$,
\begin{equation*}
I(u_n)\to c, \quad \|u_n\|\to \infty, \quad
 (1+\|u_n\|)\|I'(u_n)\|\to 0,\quad n\to\infty.
\end{equation*}
In particular,
\begin{equation}
\begin{aligned}
c&=\lim_{n\to \infty}(I(u_n)-\frac 1p\langle I'(u_n),u_n\rangle)\\
\label{2.2}
&=\lim_{n\to \infty}\Big(\Big(\frac 12-\frac 1p\big)
\int_\Omega |\nabla u_n|^2+\int_\Omega \Big(\frac 1p f(x,u_n)u_n-F(x,u_n)\Big)\Big).
\end{aligned}
\end{equation}
Let $w_n=u_n/\|u_n\|$. Since $\{w_n\}$ is bounded in $X$, up to a subsequence,
 we have as $n\to\infty$,
\begin{equation}\label{2.42}
\begin{gathered}
w_n\rightharpoonup w \quad\text{in }  X,\\
w_n\to w \quad\text{in }  L^s(\Omega),\; \ s\in[1,p^*),\\
w_n(x)\to w(x) \quad\text{ a.e. } x\in \Omega.
\end{gathered}
\end{equation}
If $w=0$, as in \cite{4}, we choose a sequence $\{t_n\}\subset [0,1]$ such that
\begin{equation*}
I(t_n u_n)=\max_{t\in [0,1]} I(tu_n).
\end{equation*}
For any $\alpha>0$, let $v_n=(2\alpha p)^{1/p}w_n$. 
Since $v_n\to 0, n\to\infty$ in $L^q(\Omega)$ and
\begin{equation*}
|F(x,t)|\leqslant C(1+|t|^q),
\end{equation*}
the continuity of  the Nemitskii operator implies that 
$F(\cdot, v_n)\to 0, n\to\infty$ in $L^1(\Omega)$.
 For $n$ large enough, since $(2\alpha p)^{1/p}\|u_n\|^{-1}\in (0,1)$,
\begin{align*}
I(t_n u_n)\geqslant I(v_n)\geqslant 2\alpha
-\int_\Omega |F(x, v_n)|\geqslant \alpha.
\end{align*}
We have shown that $I(t_n u_n)\to \infty$. Since $I(0)=0,\ I(u_n)\to c$,  
we know that $t_n\in (0,1)$ and
\begin{equation}\label{2.31}
\begin{aligned}
&\int_\Omega |\nabla(t_n u_n)|^2+\int_\Omega |\nabla(t_n u_n)|^p
-\int_\Omega f(x,t_n u_n)t_n u_n \\
&=\langle I'(t_n u_n), t_n u_n\rangle=t_n \frac{d}{dt}\big|_{t=t_n} I(t u_n)=0
\end{aligned}
\end{equation}
for $n$ large enough. Let $G(x,t)=f(x,t)t-pF(x,t)$ and
\begin{equation*}
c_1=1+\sup_{\overline{\Omega}\times[-R, R]}G(x,t)
-\inf_{\overline{\Omega}\times[-R, R]}G(x,t).
\end{equation*}
By (H5),   for any $x\in \overline{\Omega}$, $0\leqslant s\leqslant t$ or 
$t\leqslant s\leqslant 0$,
\begin{equation}\label{2.41}
G(x,s)\leqslant G(x,t)+c_1.
\end{equation}
 It follows from  \eqref{2.31} that
\begin{align*}
&\big(\frac 12-\frac 1p\big)\int_\Omega |\nabla u_n|^2
+\int_\Omega \left(\frac 1p f(x,u_n)u_n-F(x,u_n)\right)\\
&\geqslant \big(\frac 12-\frac 1p\big)\int_\Omega |\nabla t_n u_n|^2
 +\int_\Omega \left(\frac 1p f(x,t_n u_n)t_n u_n-F(x,t_n u_n)\right)
 -\frac{c_1}{p}|\Omega|\\
&= \frac 12 \int_\Omega |\nabla t_n u_n|^2
+\frac 1p \int_\Omega |\nabla t_n u_n|^p
-\int_\Omega F(x,t_n u_n)-\frac{c_1}{p}|\Omega|\\
&= I(t_n u_n)-\frac{c_1}{p}|\Omega|\to \infty, \ \ n\to\infty.
\end{align*}
This contradicts \eqref{2.2}.

If $w\neq 0$, then the set $\Theta :=\{x\in \Omega: w(x)\neq 0\}$ has 
positive Lebesuge measure. For a.e. $x\in \Theta$, we have $|u_n(x)|\to \infty$ 
as $n\to\infty$.  By (f$_2)$, as $n\to\infty$,
\begin{equation*}
\frac{F(x, u_n(x))}{|u_n(x)|^p} |w_n(x)|^p \to \infty, \quad\text{a.e. } x\in \Theta.
\end{equation*}
Using \eqref{2.1} and the Fatou lemma, we get
\begin{equation}\label{2.51}
\int_{\Theta}\frac{F(x, u_n(x))}{|u_n(x)|^p} |w_n(x)|^p \to \infty,\quad n\to\infty.
\end{equation}
The Lebegue  convergence theorem implies that $\int_{w=0} |w_n|^p\to 0$ as 
$n\to \infty$. Combining this with \eqref{2.1}, we have
\begin{equation}\label{2.5}
\int_{\Omega\setminus\Theta}\frac{F(x, u_n(x))}{|u_n(x)|^p} |w_n(x)|^p 
\geqslant -\mu\int_{\Omega\setminus\Theta} |w_n|^p\geqslant-\mu\int_{\Omega} |w_n|^p.
\end{equation}
Since $X\hookrightarrow H^1_0(\Omega)$, there exists $c_2>0$ such that
\begin{equation*}
\|u_n\|_H^2\leqslant c_2\|u_n\|^2.
\end{equation*}
It follows  that
\begin{equation*}
\frac {c_2}{2} \|u_n\|^2+\frac 1p \|u_n\|^p-c
\geqslant \frac 12\|u_n\|_H^2+\frac 1p \|u_n\|^p-c=\int_\Omega F(x, u_n)+o(1).
\end{equation*}
By \eqref{2.51}, \eqref{2.5} and \eqref{2.42}, we have
\begin{align*}
\frac {c_2}{2} \|u_n\|^{2-p}+\frac 1p -\frac{c}{\|u_n\|^p}
&\geqslant \int_\Omega \frac{F(x, u_n)}{\|u_n\|^p}+o(1) \\
&=\Big(\int_{\Theta}+\int_{\Omega\setminus\Theta}\Big)
\frac{F(x, u_n(x))}{|u_n(x)|^p} |w_n(x)|^p+o(1)\to \infty, 
\end{align*}
as $n\to\infty$. This is impossible.

Secondly, up to a subsequence, we may assume that
\begin{gather*}
u_n\rightharpoonup u_0 \quad \text{in } X,\\
u_n\to u_0 \quad\text{in }  L^s(\Omega),\quad s\in[1,p^*),\\
u_n(x)\to u_0(x) \quad\text{a.e. } x\in  \Omega.
\end{gather*}
We will prove $u_n\to u_0$ in $X$. Defined $A:X\to X^*$ by
\[
\langle Au, v \rangle 
=\int_\Omega \nabla u\cdot \nabla v
+\int_\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla v,\quad u,v\in X.
\]
For any $u, v\in X$, there exists $c_3>0$ such that (see \cite{3})
\begin{equation} \label{2.7}
\begin{aligned}
&\langle Au_n-Au_0, u_n-u_0 \rangle \\
&= \int_\Omega |\nabla(u_n-u_0)|^2
 +\int_\Omega(|\nabla u_n|^{p-2}\nabla u_n-|\nabla u_0|^{p-2}\nabla u_0)
 \cdot\nabla(u_n-u_0)\\
&\geqslant \int_\Omega |\nabla(u_n-u_0)|^2+c_3\int_\Omega |\nabla(u_n-u_0)|^p\\
&\geqslant c_3 \|u_n-u_0\|^p.
\end{aligned}
\end{equation}
By (H3) we know that
\begin{equation*}
\int_\Omega f(x, u_n)(u_n-u_0)\to 0, \quad  n\to\infty.
\end{equation*}
According to $\langle I'(u_n), u_n-u_0\rangle\to 0$ as $n\to\infty$, 
we have $\langle Au_n, u_n-u_0\rangle\to 0$ as $n\to\infty$. Then
\begin{equation*}
\langle Au_n-Au_0, u_n-u_0 \rangle\to 0,\quad  n\to\infty.
\end{equation*}
It follows form \eqref{2.7} that $u_n\to u_0$ as $n\to\infty$.
The proof is complete.
\end{proof}

For the proof of Theorem {thm1.1}, we assume that $I$ has only finitely many 
critical points. Since $I$ satisfies the Cerami condition, the critical
 group $C_k(I, \infty), k\in\mathbb{N}$ makes sense.

\begin{lemma}\label{lem2.2}
Assume that {\rm (H3)-(H6)} hold. Then for any $k\in \mathbb{N}$, we have 
$C_k(I,\infty)\cong 0$.
\end{lemma}

\begin{proof}
Let $S=\{u\in X: \|u\|=1\}$. By (H4), we see that for any $u\in S$,
\begin{equation*}
I(tu)\to -\infty,\quad t\to \infty.
\end{equation*}
Choose $a<\min\{\inf_{\|u\|\leqslant2}I(u),-\frac 1p c_1|\Omega|\}$.
Then for any $u\in S$, there exists $t>1$ such that $I(tu)\leqslant a$, that is,
\begin{equation*}
I(tu)=\frac {t^2}2\int_\Omega |\nabla u|^2+\frac {t^p}p
-\int_\Omega F(x,tu)\leqslant a.
\end{equation*}
By \eqref{2.41}, we can find that $G(x,t)\geqslant -c_1$ when $s=0$. Therefore,
\begin{align*}
\frac d{dt}I(tu)
&= t\int_\Omega |\nabla u|^2+t^{p-1}-\int_\Omega uf(x,tu)\\
&\leqslant \frac 1t\Big(pa+\int_\Omega pF(x,tu)-\int_\Omega tu f(x,tu)\Big)\\
&=\frac 1t\Big(pa-\int_\Omega G(x,tu)\Big)\\
&\leqslant \frac 1t(pa+c_1|\Omega|)<0.
\end{align*}
By  the implicit function theorem, there exists an unique $T\in C(S,\mathbb{R})$ 
such that for any $u\in S$, $I(T(u)u)=a$.

For $u\neq 0$, set $\widetilde{T}(u)=\frac 1{\|u\|}T(\frac {u}{\|u\|})$. 
Then $\widetilde{T}\in C(X\setminus\{\theta\},\mathbb{R})$ and for all 
$u\in X\setminus\{\theta\}$, $I(\widetilde{T}(u)u)=a$. Moreover, if $I(u)=a$, 
then $\widetilde{T}(u)=1$.

We define a function $\widehat{T}: X\setminus\{\theta\}\to \mathbb{R}$ as
\begin{equation*}
\widehat{T}(u)=\begin{cases}
\widetilde{T}(u),&I(u)> a,\\
1,&I(u)\leqslant a.
\end{cases}
\end{equation*}
Since $I(u)=a$ implies $\widetilde{T}(u)=1$, we know that 
$\widehat{T}\in C(X\backslash\{\theta\},\mathbb{R})$.

Finally, we set $\eta:[0,1]\times(X\setminus\{\theta\})\to X\setminus\{\theta\}$ as
\begin{equation*}
\eta(s,u)=(1-s)u+s\widehat{T}(u)u.
\end{equation*}
It is easy to see that $\eta$ is a strong deformation retract from 
$X\setminus \{\theta\}$ to $I^a$. Thus,
\begin{equation*}
C_k(I,\infty)=H_k(X,I^a)\cong H_k(X,X\setminus\{\theta\})=0,\quad k\in\mathbb{N}.
\end{equation*}
Here,  $H_k(A, B), k\in\mathbb{N}$ denotes the kth
singular relative homology group of the topological pair $(A, B)$ with 
coefficients in a field $\mathbb{F}$. The proof is complete.
\end{proof}

Since $l>\lambda_1$ and $l\not\in\sigma(\Delta)$, there exists 
$n_0\in\mathbb{N}$ such that 
$\lambda_1<\lambda_2\leqslant\dots\leqslant\lambda_{n_0}<l
<\lambda_{n_0+1}\leqslant\dots$. Let $\varphi_i$ be
the corresponding eigenfunction of $\lambda_i$ with $\|\varphi_i\|=1$ 
and $V=\operatorname{span}\{\varphi_1, \varphi_2\dots \varphi_{n_0}\}$.
Assume $W$ is the complementary space of $V$ in $X$. 
For the details on the term of complementary space we refer the readers 
to \cite[p.94]{j}.  Then $X=V\bigoplus W$. We may assume that
\begin{equation}\label{3.40}
\int_\Omega|\nabla u|^2\geqslant \lambda_{n_0+1}\int_\Omega u^2,\quad u\in W.
\end{equation}
In fact, since $\partial\Omega$ is smooth enough, 
$\operatorname{span}\{\varphi_i:i=1,2,\dots\}$ is dense in $H^m_\vartheta(\Omega)$ 
with some $m\in\mathbb{N}$ and $2m-2\geqslant N$, where
 \begin{equation*}
 H^m_\vartheta(\Omega)=\big\{u\in H^m(\Omega):\Delta^ju=0\quad\text{on }
\partial\Omega \text{ for } j<\frac{m}{2}\big\}
 \end{equation*}
 denotes the Hilbert space with the scalar product
 \begin{equation*}
 (u,v):=\begin{cases}
 \int_\Omega\Delta^ku\Delta^kv,&m=2k,\\[4pt]
 \int_\Omega\nabla(\Delta ^ku)\cdot\nabla(\Delta^kv),&m=2k+1.
\end{cases}
 \end{equation*}
 $H^m_\vartheta(\Omega)$ is a closed subspace of $H^m(\Omega)$ which satisfies 
$H^m_\vartheta(\Omega)\subset H^m(\Omega)$ with continuous embedding; 
see \cite{ga} for the details.  Since the embedding 
$H^m(\Omega)\hookrightarrow W^{1,p}(\Omega)$ is continuous, we have that 
$\operatorname{span}\{\varphi_i:i=1,2,\dots\}$ is dense in $W^{1,p}_0(\Omega)$. 
Let $W=\overline{\operatorname{span}\{\varphi_i:i=n_0+1, n_0+2, \dots\}}$,
 the closure of $\operatorname{span}\{\varphi_i:i=n_0+1, n_0+2, \dots\}$ in $X$. 
Then the continuous embedding $W^{1,p}_0(\Omega)\hookrightarrow H^1_0(\Omega)$ 
implies the desired result \eqref{3.40}.

\begin{lemma}\label{lem2.3}
Assume that {\rm (H3)--(H6)} are satisfied. Then there exists $\rho>0$ such that
\begin{equation}\label{2.70}
\begin{gathered}
I(u)\leqslant 0, \quad u\in V,\quad \|u\|\leqslant \rho,\\
I(u)>0,\quad u\in W,\quad 0<\|u\|\leqslant \rho.
\end{gathered}
\end{equation}
\end{lemma}

\begin{proof}
Let $c_4=\min_{1\leqslant i\leqslant n_0}\int_\Omega\varphi_i^2$. 
For $u=\sum_{i=1}^{n_0}a_i\varphi_i\in V$, define
\begin{equation*}
|u|_1^2=\sum_{i=1}^{n_0} a_i^2, \quad
|u|_p^p=\sum_{i=1}^{n_0} |a_i|^p, \quad
|u|_\infty=\max_{x\in\Omega}|u(x)|.
\end{equation*}
We know that there exist $c_5, c_6, c_7>0$ such that
\begin{equation*}
\|u\|^2\leqslant c_5|u|_1^2, \quad 
|u|_p^p\leqslant c_6\|u\|^p, \quad
|u|_\infty\leqslant c_7\|u\|.
\end{equation*}
It follows from (H6) that there exist $\varepsilon, r>0$ such that 
$\lambda_{n_0}+2\varepsilon<l<\lambda_{n_0+1}-\varepsilon$ and
\begin{equation}\label{3.41}
\frac{(l-\varepsilon)t^2}{2}\leqslant F(x,t)\leqslant\frac{(l+\varepsilon)t^2}{2},\quad
x\in\Omega, \; |t|\leqslant r.
\end{equation}
Fix $0<\rho_1\leqslant\min\{r/c_7, (\varepsilon p c_4/(2 c_5 c_6))^{1/(p-2)}\}$. 
For $u=\sum_{i=1}^{n_0} a_i\varphi_i\in V$ with $\|u\|\leqslant \rho_1$, we have
\begin{align*}
I(u)
&= \frac 12\int_\Omega |\nabla u|^2+\frac 1p\int_\Omega |\nabla u|^p-\int_\Omega F(x, u)\\
&\leqslant \frac{1}{2}\sum_{i=1}^{n_0} a_i^2\lambda_i\int_\Omega\varphi_i^2+\frac{1}{p}\sum_{i=1}^{n_0}|a_i|^p-\frac{\lambda_{n_0}+\varepsilon}{2}\sum_{i=1}^{n_0} a_i^2\int_\Omega\varphi_i^2\\
&\leqslant -\frac {\varepsilon }2c_4\sum_{i=1}^{n_0} a_i^2+\frac{1}{p}\sum_{i=1}^{n_0}|a_i|^p\\
&\leqslant -\frac {\varepsilon }2\frac{c_4}{c_5}\|u\|^2+\frac{1}{p}c_6\|u\|^p
\leqslant 0.
\end{align*}
By (H3), \eqref{3.40} and \eqref{3.41},  we have the following estimates for
 $u\in W$.
\begin{align*}
I(u)
&= \frac 12\int_\Omega |\nabla u|^2+\frac 1p\int_\Omega |\nabla u|^p
 -\int_\Omega F(x, u)\\
&= \frac 12\int_\Omega \left(|\nabla u|^2-\lambda_{n_0+1}|u|^2\right)
+\frac 1p\int_\Omega |\nabla u|^p\\
&\quad -\int_{|u|\leqslant r} \Big(F(x, u)-\frac {\lambda_{n_0+1} }2|u|^2\Big)
-\int_{|u|\geqslant r} \Big(F(x, u)-\frac {\lambda_{n_0+1}}2|u|^2\Big)\\
&\geqslant \frac 1p\int_\Omega |\nabla u|^p-c_8\int_\Omega|u|^q\\
&\geqslant \frac 1p\|u\|^p-c_9\|u\|^q.
\end{align*}
Then, there exists $\rho_2>0$ such that
\begin{equation*}
I(u)>0, \quad u\in W,\quad 0<\|u\|\leqslant\rho_2.
\end{equation*}
Taking $\rho=\min\{\rho_1, \rho_2\}$,  the proof is complete.
\end{proof}

Combining  Lemma \ref{lem2.3} with the linking theorem at zero \cite{m}, we can
 get the following lemma.

\begin{lemma}\label{lem2.4}
Assume that {\rm (H3)--(H6)} holds. Then $\theta$, the zero function of $X$, 
is a critical point of $I$ and $C_{n_0}(I,\theta)\not\cong0$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
Since $f(x,0)=0$, the zero function $\theta$ is a trivial  critical point of $I$.
According to Lemma \ref{lem2.4}, we have $C_{n_0}(I,\theta)\not\cong 0$ while by 
Lemma \ref{lem2.2} $C_k(I, \infty)\cong0$ for all $k\in \mathbb{N}$. Now the desired result 
of Theorem \ref{thm1.1} follows from Theorem \ref{thm1.2}.
\end{proof}

We conclude this article with some remarks. We obtained the existence of nontrivial  
solutions for a type of (2,$p$)-Laplacian equations with  the nonlinearity $f$ 
having $(p-1)$-superlinear and subcritical growth   by critical point methods 
and Morse theory. In connection with the study, a natural question is what 
happens if the $(2,p)$-Laplacian is replaced by  the general $(q,p)$-Laplacian 
with $1<q<p$.  In  our proof of the main result, the main ingredient  is the 
decomposition  $X=V\oplus W$ with the property
\begin{equation*}
\int_\Omega|\nabla u|^2\geqslant \lambda_{n_0+1}\int_\Omega u^2,\quad u\in W,
\end{equation*}
which is deduced  from the  properties of the $-\Delta$'s eigenfunctions  and 
some progresses for the polyharmonic equations with Navier boundary conditions. 
Recently,  some researches have been obtained for the $p$-Laplacian type equation 
by Morse theory; see \cite{perera} for example. We think the question mentioned 
above deserves a further investigation with the developments of the spectrum 
theory of $-\Delta_p$ and some ideas form the researches of the $p$-Laplacian 
type equation by Morse theory.

\subsection*{Acknowledgments}
This research was partially supported by National Natural Science Foundation
of China (11571209, 11671239, 11671026), Science Council of Shanxi Province
 (2014021009-1, 2015021007),
and Special Foundation of Beijing Educational Committee (KZ201510028032).

The authors sincerely thank the reviewers for their valuable suggestions 
and useful comments.

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\end{document}