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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 139, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/139\hfil Semilinear elliptic equations]
{Semilinear elliptic equations with dependence on the gradient}

\author[G. Liu, S. Shi, Y. Wei \hfil EJDE-2012/139\hfilneg]
{Guanggang Liu, Shaoyun Shi, Yucheng Wei}  % in alphabetical order

\address{Guanggang Liu \newline
College of Mathematics, Jilin University,
Changchun 130012, China}
\email{lgg112@163.com}

\address{Shaoyun Shi \newline
College of Mathematics, Jilin University,
Changchun 130012,  China}
\email{shisy@jlu.edu.cn}

\address{Yucheng Wei \newline
College of Mathematics, Jilin University,
 Changchun 130012,  China\newline
Department of Mathematics,  Hechi University,
Yizhou 546300, China}
\email{ychengwei@126.com}

\thanks{Submitted April 17, 2012. Published August 19, 2012.}
\subjclass[2000]{35J20, 35J25, 35J60}
\keywords{Semilinear equation; Morse theory; critical group; iterative method}

\begin{abstract}
 In this article we consider elliptic equations whose nonlinear term
 depends on the gradient of the unknown. We assume that the nonlinearity  
 has a asymptotically linear growth at zero and at infinity
 with respect to the second variable. By applying Morse theory  and
 an iterative method, we prove the existence of nontrivial solutions.
\end{abstract}

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


\section{Introduction}

In this article we consider the following elliptic equation with
 dependence on the gradient,
 \begin{equation}\label{cc}
\begin{gathered}
-\Delta u=f(x,u,\nabla u) , \quad \text{in }\Omega,\\
u=0, \quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^N$ $(N\geq3)$ is a bounded domain with
smooth boundary. Since the nonlinearity $f$ depends on the gradient
of the solution, solving \eqref{cc} is not variational. In fact
 the well developed critical point theory cannot be applied
directly.
There have been several works on this equation, using sub and
supersolution, topological degree, fixed point theorems and Galerkin
method; see, for instance,
\cite{Alves, Amann,Figueiredo2,Pohozaev,Wang,Xavier,Yan}.

In \cite{Figueiredo1}, de Figueiredo,  Girardi  and  Matzeu
developed a quite different  method of variational type. Under the
assumptions that $f$ has a superlinear subcritical growth at zero
and at infinity with respect to the second variable, they obtained the
existence of a positive and a negative solutions of
\eqref{cc} by using the mountain pass theorem and iterative
technique.  Later, this method was applied to quasilinear elliptic
equations \cite{Giovany,Girardi1,Servadei}, Hamiltonian
systems \cite{Girardi2} and impulsive differential
equations \cite{Teng}.

In general the above papers which used mountain pass technique
assume that the nonlinearity has a superlinear subcritical growth 
at zero and at infinity with respect to the second variable. 
Here we show that Morse theory and iterative method can be
used to find solutions to \eqref{cc} under the assumption
that  $f$ has a asymptotically linear growth at zero and at infinity
with respect to the second variable.

Let
$0<\lambda_1<\lambda_2\leq\lambda_3\leq\dots \leq\lambda_{k}
\leq\dots $
be the  eigenvalues associated with the eigenvectors
 $\varphi_1,
\varphi_2, \varphi_3, \varphi_4,\dots $ of
$-\Delta$ with Dirichlet boundary condition, and we make the
following assumptions:
\begin{itemize}

\item[(H0)] $f: \overline{\Omega}\times \mathbb{R}\times
\mathbb{R}^N\to \mathbb{R}$ is continuous;

\item[(H1)] $f(x,t,\xi)=\lambda t+g_0(x,t,\xi)$, where
$\lambda_{j}<\lambda<\lambda_{j+1}, g_0(x,t,\xi)=o(|t|)$ as
$t\to 0$ uniformly for $x\in \overline{\Omega}, \xi\in \mathbb{R}^N$ ;


\item[(H2)] $f(x,t,\xi)=\mu t+g(x,t,\xi)$, where
$\lambda_{k}<\mu<\lambda_{k+1}, k\neq j, k\geq1, g(x,t,\xi)=o(|t|)$
as $|t|\to\infty$ uniformly for $x\in \overline{\Omega},
\xi\in \mathbb{R}^N$;

\item[(H3)] $f(x,t,\xi)=\mu t+g(x,t,\xi)$, where
$\mu=\lambda_{k}$ and
$\lambda_{k-l-1}<\lambda_{k-l}=\lambda_{k-l+1}=\dots
 =\lambda_{k-1}=\lambda_{k}<\lambda_{k+1}$,
$k\neq j$, $k\geq l+1$, $|g(x,t,\xi)|\leq C$ and
$G(x,t,\xi)\to -\infty$ as $|t|\to\infty$ uniformly
for $x\in \overline{\Omega}$, $\xi\in \mathbb{R}^N$, where $C>0$ is a
constant, $G(x,t,\xi)=\int_0^{t}g(x,s,\xi)ds$;


\item[(H4)] For any
$x\in\overline{\Omega}$, $t_1,t_2\in \mathbb{R}$, 
$\xi_1, \xi_2\in \mathbb{R}^N$, $f(x,t,\xi)$ satisfies the 
Lipschitz condition
$$
|f(x,t_2,\xi_2)-f(x,t_1,\xi_1)| \leq L(|t_2-t_1|+|\xi_2-\xi_1|),
$$
where $L>0$ is a constant.
\end{itemize}

By (H1), zero is a solution of \eqref{cc}, called trivial
solution. The purpose of this article is to find nontrivial solutions.
Our main results as as follows:

\begin{theorem}\label{thm1.1}
Assume that {\rm (H0), (H1), (H2), (H4)} hold. If
 $0<\frac{L\sqrt{\lambda_1}}{\lambda_1-L}<1$, then
\eqref{cc} has at least a nontrivial weak solution.
\end{theorem}

\begin{theorem}\label{thm1.2}
Assume that {\rm (H0), (H1), (H3), (H4)} hold. If
 $0<\frac{L\sqrt{\lambda_1}}{\lambda_1-L}<1$, then
\eqref{cc} has at least a nontrivial weak solution.
\end{theorem}

This article is organized as follows. 
In section 2 we give a simple revisit to Morse theory. 
In section 3 we prove Theorem \ref{thm1.1} and Theorem \ref{thm1.2} 
by using Morse theory and iterative method. An
example will be given in section 4.


\section{Preliminaries about Morse theory}\label{prelim}

Let $H$ be a real Hilbert space and $J\in C^{1}(H,\mathbb{R})$ be a
functional satisfying the (PS) condition. Denote by $H_q(A,B)$ the
$q$-th singular relative homology group of the topological pair with
coefficients in a field $G$. Let $u$ be an isolated critical point
of $J$ with $J(u)=c$. The group
$$
C_q(J,u):=H_q(J^{c},J^{c}\setminus\{u\}),\quad  q\in \mathbb{Z},
$$
is called the $q$-th critical group of $J$ at $u$, where
$J^{c}=\{u\in H\mid J(u)\leq c\}$. Denote 
$K=\{u\in H \setminus J'(u)=0\}$. Assume that $K$ is a finite set. 
Take $a<\inf J(K)$. The critical groups of $J$ at infinity are defined by
$$
C_q(J,\infty):=H_q(H,J^{a}\setminus\{u\}),\quad  q\in \mathbb{Z}.
$$
The following result is important in proving the existence of
nontrivial critical points.

\begin{proposition}[{\cite[Proposition 3.6]{Bar}}] \label{prop1}
Suppose $J$ satisfies the (PS) condition. If $K=\emptyset$, then
$C_q(J,\infty)\cong 0$,  $q\in \mathbb{Z}$.
If $K=\{u_0\}$, then
$C_q(J,\infty)\cong C_q(J, u_0)$,  $q\in \mathbb{Z}$.
\end{proposition}



 Let $A_{\infty}$ and $A_0$ be bounded self-adjoint operators defined on $H$.
According to their spectral decomposition, $H=H^{+}\oplus
H^0\oplus H^{-}$, where $H^{+}$, $H^0$, $H^{-}$ are invariant
subspaces corresponding to the positive, zero and negative spectrum
of $A_{\infty}$, respectively, similarly, $H=H_0^{+}\oplus
H_0^0\oplus H_0^{-}$, where $H_0^{+}$, $H_0^0$,
$H_0^{-}$ are invariant subspaces corresponding to the positive,
zero and negative spectrum of $A_0$, respectively. 
Let $P_0:H\to H^0$ be the orthogonal projector.

  Consider the functionals
$$
\Phi(u)=\frac{1}{2}\langle A_{\infty}u,u\rangle+\varphi(u),\quad
\Phi_0(u)=\frac{1}{2}\langle A_0u,u\rangle+\varphi_0(u).
$$
We make the following assumptions:
\begin{itemize}

\item[(A1)] $(A_{\infty})_{\pm}:=A_{\infty}|_{H^{\pm}}$ has a bounded
inverse on $H^{\pm}$.

\item[(A2)] $\gamma:=\dim(H^{-}\oplus H^0)<\infty$.

\item[(A3)] $\varphi\in C^{1}(H,\mathbb{R})$ has a  compact gradient
mapping $\nabla\varphi(u)$, and $\nabla\varphi(u)=o(\|u\|)$ as
$\|u\|\to\infty$. In addition, if $\operatorname{dim}H_0\neq0$, we assume
$$
\|\nabla\varphi(u)\|\leq C, \forall u\in H, \quad
\varphi(P_0u)\to-\infty \quad\text{as } \|P_0u\|\to\infty.
$$

\item[(A4)] $(A_0)_{\pm}:=(A_0)|_{H_0^{\pm}}$ has a bounded
inverse on $H_0^{\pm}$.

\item[(A5)] $\beta:=\dim(H_0^{-})<\infty$, and $\operatorname{dim}H_0^0=0$.

\item[(A6)] $\varphi_0\in C^{1}(H,\mathbb{R})$ has a  compact
gradient mapping $\nabla\varphi_0(u)$, and
$$
\nabla\varphi_0(u)=o(\|u\|)\quad\text{as } \|u\|\to0.
$$
\end{itemize}

Also we use the following results.

\begin{theorem}[{\cite[Lemma 5.1]{Cha}}]\label{thm2.2}
Assume that {\rm (A1)--(A3)} hold, then  $\Phi$ satisfies the (PS)
condition, and
\[
 C_q(\Phi, \infty)=\begin{cases}
G,  &q=\gamma, \\
0,  &q\neq \gamma.
\end{cases}
\]
\end{theorem}

\begin{theorem}[{\cite[Theorem 4.1]{Cha}}]\label{thm2.3}
Assume that {\rm (A4)--(A6)} hold, then
\[ 
C_q(\Phi_0, 0)=\begin{cases}
G,  &q=\beta, \\
0,  &q\neq \beta.
\end{cases}
\]
\end{theorem}



\section{Proof of Theorems \ref{thm1.1} and  \ref{thm1.2}}
\label{proof}

Let $H_0^{1}(\Omega)$ be the usual Sobolev space with the inner
product 
$$
\langle u,v\rangle=\int_{\Omega}\nabla u(x)\cdot\nabla v(x)\,dx,  \quad
 \forall u, v\in H_0^{1}(\Omega).
$$
For  $w\in H_0^{1}(\Omega)$, consider the problem
\begin{equation}\label{ccccc}
\begin{gathered}
-\Delta u=f(x,u,\nabla w), \quad \text{in }\Omega,\\
u=0, \quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
and the associated functional 
$I_w: H_0^{1}(\Omega)\to \mathbb{R}$,
$$
I_w(v)=\frac{1}{2}\int_{\Omega}|\nabla v(x)|^{2}\,dx
 -\int_{\Omega}F(x,v(x),\nabla w(x))\,dx.
$$
By (H0)  (H2) or (H0) (H3), $I_w\in C^{1}(H_0^{1}(\Omega),\mathbb{R})$, and
 the weak solutions of the problem \eqref{ccccc}
corresponds to the critical points of the functional $I_w$, see
\cite{Rab}.

Define the operators $L_{\infty}, L_0:
H_0^{1}(\Omega)\to H_0^{1}(\Omega)$  by
$L_{\infty}u=u-\mu(-\triangle)^{-1}u$ and
$L_0u=u-\lambda(-\triangle)^{-1}u$.
 Obviously, $L_{\infty}$ and $L_0$ are bounded
self-adjoint operators. Let
$$
\phi_w(u)=\int_{\Omega}g(x,u(x),\nabla w(x))\,dx,\quad
\phi_{w0}(u)=\int_{\Omega}g_0(x,u(x),\nabla w(x))\,dx.
$$ 
It is well known that $\nabla \phi_w, \nabla \phi_{w0}$ are compact mappings.
By (H1), (H2) or (H1), (H3), $\nabla
\phi_w(u)=o(\|u\|)$ as $\|u\|\to\infty$, and $\nabla
\phi_{w0}=o(\|u\|)$ as $\|u\|\to0$. We can rewrite the
functional $I_w$ by
$$
I_w(u)=\frac{1}{2}\langle L_{\infty}u,u \rangle-\phi_w(u)=
\frac{1}{2}\langle L_0u,u \rangle-\phi_{w0}(u).
$$
According to the spectral decomposition of the operator
$L_{\infty}$, $H_0^{1}(\Omega)=H^{+}\oplus H^0\oplus H^{-}$,
where $H^{+}, H^0, H^{-}$ are invariant subspaces corresponding to
the positive, zero and negative spectrum of $L_{\infty}$
respectively.

If (H2) holds, then 
$H^{-}=\operatorname{span}\{\varphi_1,\varphi_2,\dots \varphi_{k}\}$,
$H^{+}=(H^{-})^{\perp}$. 

If (H3) holds, then $H^{-}=\operatorname{span}\{\varphi_1,\varphi_2,\dots 
\varphi_{k-l-1}\}$,
$H^0=\operatorname{span}\{\varphi_{k-l}, \dots \varphi_{k}\}$,
$H^{+}=(H^0\oplus H^{-})^{\perp}$.

Similarly, according to the spectral decomposition of the operator
$L_0$, $H_0^{1}(\Omega)=H_0^{+}\oplus H_0^0\oplus
H_0^{-}$, where $H_0^{+}, H_0^0, H_0^{-}$ are invariant
subspaces corresponding to the positive zero and negative spectrum
of $L_0$ respectively. If (H1) holds, then $L_0$ is invertible
and $H_0^{-}=\operatorname{span}\{\varphi_1,\varphi_2,\dots \varphi_{j}\}$,
$H_0^{+}=(H_0^{-})^{\perp}$.


\begin{lemma}\label{l1}
Assume that {\rm (H0)--(H2)} hold.  Then for any $w\in
H_0^{1}(\Omega)$, \eqref{ccccc} has at least a nontrivial weak
solution.
\end{lemma}

\begin{proof} 
By (H2), $\operatorname{dim}H^0=0$,
$L_{\infty}|_{H^{\pm}}$ has a bounded inverse on $H^{\pm}$ and
$\operatorname{dim}H^{-}=k$, thus  (A1), (A2) and (A3) hold. So by Theorem
\ref{thm2.2}, $I_w$ satisfies (PS) condition and
\[ 
C_q(I_w, \infty)=\begin{cases}
G,  &q=k, \\
0,  &q\neq k.
\end{cases}
\]
By (H1),  $\operatorname{dim}H^0=0$, $L_0|_{H_0^{\pm}}$ has a bounded inverse
on $H_0^{\pm}$ and $\operatorname{dim}H^{-}=j$, thus (A4), (A5) and (A6) hold.
 By Theorem \ref{thm2.3}
\[ C_q(I_w, 0)=\begin{cases}
G,  &q=j, \\
0,  &q\neq j.
\end{cases}
\]
Since $k\neq j$, $C_q(I_w, \infty)\neq C_q(I_w, 0)$ for some
$q\in\mathbb{Z}$, hence by Proposition \ref{prop1}, $I_w$ has at
least a nontrivial critical point and \eqref{ccccc} has at least a
nontrivial weak solution. 
\end{proof}

\begin{lemma}\label{l3}
Assume that {\rm (H0), (H1), (H3)} hold.  Then for any 
$w\in H_0^{1}(\Omega)$, \eqref{ccccc} has at least a nontrivial weak
solution.
\end{lemma}

\begin{proof} By (H3), $L_{\infty}|_{H^{\pm}}$
has a bounded inverse on $H^{\pm}$, and $\operatorname{dim}(H^{-}\oplus H^0)=k$,
so (A1), (A2) hold. On the other hand, $\operatorname{dim}H^0>0$, but it can be
checked that $\|\nabla \phi_w(u)\|\leq C'$ for any 
$u\in H_0^{1}(\Omega)$ and a constant $C'>0$, and
$\phi_w(u)\to-\infty$ with $u\in H^0$ as
$\|u\|\to\infty$. Indeed, by (H3), H\"older
inequality and Sobolev inequality, for any $u,v\in
H_0^{1}(\Omega)$, we have
$$
| \langle\nabla \phi_w(u),v\rangle| \leq
\int_{\Omega}|g(x,u,\nabla w)||v| \,dx \leq
C(\int_{\Omega}|v|^{2})^{1/2} \leq C'\|v\|,
$$
where $C,C'>0$ are constants, this implies 
$\|\nabla \phi_w(u)\|\leq C'$ for all $u\in H_0^{1}(\Omega)$.

We claim that $\phi_w(u)\to-\infty$ with $u\in H^0$
 as  $\|u\|\to\infty$. If this is not true,
  then there exists a sequence $\{u_{n}\}$ and constant $M>0$ such that
$u_{n}\in H^0$, $\|u_{n}\|\to\infty$ as
$n\to\infty$, and $\phi_w(u_{n})\geq-M$. 
Let $\tilde{u}_{n}=\frac{u_{n}}{\|u_{n}\|}$, then 
$\tilde{u}_{n}\in H^0$ and $\|\tilde{u}_{n}\|=1$.
 By $\dim H^0<\infty$, there exists a subsequence of 
$\{\tilde{u}_{n}\}$ still denoted by  $\{\tilde{u}_{n}\}$,
 and $\tilde{u}$ such that $\tilde{u}_{n}$
 converges strongly to $\tilde{u}\in H^0$ as $n\to\infty$,
then $\tilde{u}$ satisfies the equation
\begin{equation}
\begin{gathered}
-\Delta \tilde{u}=\lambda_{k}\tilde{u} , \quad \text{in }\Omega,\\
\tilde{u}=0, \quad \text{on }\partial\Omega.
\end{gathered}
\end{equation}
Since $\tilde{u}\neq0$, by the unique continuation property as in
\cite{Heinz}, $\tilde{u}\neq0$ a.e. in $\Omega$, which implies
$u_{n}\to\infty$ a.e. in $\Omega$. Hence by (H3),
$G(x,u_{n}(x),\nabla w(x))\to-\infty$ a.e. in $\Omega$, then
$$
\phi_w(u_{n})=\int_{\Omega}G(x,u_{n}(x),\nabla w(x)) \,dx\to-\infty
$$ 
as $n\to\infty$,   we obtain a contradiction.
Therefore, (A3) holds.

 Next by using the argument used in the proof of Lemma \ref{l1} we
 complete the proof.
\end{proof}

\begin{lemma}\label{l4}
There exists a constant $c_1>0$  independent of $w$ such that
$\|u_w\|\geq c_1$  for all solutions $u_w$ obtained in Lemma
\ref{l1} or Lemma \ref{l3}.
\end{lemma}

\begin{proof} First we decompose $u_w$ as
$u_w=u_w^{+}+u_w^{-}\in H_0^{+}\oplus H_0^{-}$. Since
$u_w$ is a weak solution of the problem \eqref{ccccc}, one has
\begin{equation}\label{e2}
 \int_{\Omega}\nabla
u_w\cdot \nabla \phi \,dx =\int_{\Omega}(\lambda
u_w+g_0(x,u_w,\nabla w))\phi \,dx, \quad
\forall \phi\in H_0^{1}(\Omega).
\end{equation}
Particularly, take $\phi=u_w^{+}-u_w^{-}$ into \eqref{e2}, we
have
\begin{equation}\label{a}
\int_{\Omega}\nabla u_w\cdot\nabla (u_w^{+}-u_w^{-})-\lambda
u_w(u_w^{+}-u_w^{-}) \,dx
=\int_{\Omega}g_0(x,u_w,\nabla w)(u_w^{+}-u_w^{-})\,dx.
\end{equation}
 By (H1),
$\lambda_{j-1}<\lambda<\lambda_{j}$, then we have
\begin{equation}
\begin{split}
&\int_{\Omega}\nabla u_w(x)\cdot\nabla
(u_w^{+}-u_w^{-})-\lambda u_w(x)(u_w^{+}-u_w^{-}) \,dx \\
&= \int_{\Omega}(|\nabla u_w^{+}|^{2}-\lambda |u_w^{+}|^{2})
-(|\nabla u_w^{-}|^{2}-\lambda |u_w^{-}|^{2})\,dx \\
&\geq (1-\frac{\lambda}{\lambda_{j}})\int_{\Omega}|\nabla
u_w^{+}|^{2}\,dx
+(\frac{\lambda}{\lambda_{j-1}}-1)\int_{\Omega}|\nabla
u_w^{-}|^{2}\,dx\\
&\geq m\int_{\Omega}|\nabla
u_w|^{2}\,dx,
\end{split}\label{aa}
\end{equation}
where $m=\min\{(1-\frac{\lambda}{\lambda_{j}}),
(\frac{\lambda}{\lambda_{j-1}}-1)\}>0$.
Fix  $(N+2)/(N-2)>p>1$, by (H1) (H2) or (H1) (H3), for  any $\epsilon>0$, there exists constant
$k_{\epsilon}>0$ such that
$|g_0(x,t,\xi)|\leq\epsilon|t|+k_{\epsilon}|t|^{p}$. By
H\"older inequality and Sobolev  inequality
\begin{equation}
\begin{split}
&\int_{\Omega}g_0(x,u_w(x),\nabla w(x))(u_w^{+}-u_w^{-})\,dx
 \\
&\leq \int_{\Omega}(\epsilon|u_w(x)|+k_{\epsilon}|u_w(x)|^{p})(|u_w^{+}|+|u_w^{-}|)\,dx  \\
&\leq \epsilon\|u_w\|_{L^{2}(\Omega)}\|u_w^{+}\|_{L^{2}(\Omega)}
+\epsilon\|u_w\|_{L^{2}(\Omega)}\|u_w^{-}\|_{L^{2}(\Omega)} \\
&\quad +k_{\epsilon}\|u_w\|_{L^{p+1}(\Omega)}^{p}\|u_w^{+}\|_{L^{p+1}(\Omega)}
+k_{\epsilon}\|u_w\|_{L^{p+1}(\Omega)}^{p}\|u_w^{-}\|_{L^{p+1}(\Omega)}\\
&\leq \frac{\epsilon}{\lambda_1}\|u_w\|\|u_w^{+}\|+\frac{\epsilon}{\lambda_1}\|u_w\|\|u_w^{-}\|
+Ck_{\epsilon}\|u_w\|^{p}\|u_w^{+}\|
+Ck_{\epsilon}\|u_w\|^{p}\|u_w^{-}\|\\
&\leq \frac{2\epsilon}{\lambda_1}\|u_w\|^{2}+2Ck_{\epsilon}\|u_w\|^{p+1}.
\end{split} \label{aaa}
\end{equation}
Combining \eqref{a}, \eqref{aa} and \eqref{aaa} we obtain
$(m-\frac{2\epsilon}{\lambda_1})\| u_w(x)\|^{2} \leq
2Ck_{\epsilon}\| u_w(x)\|^{p+1}$.
Since $m>0$, we can take $\epsilon>0$ sufficiently small such that
$m-\frac{2\epsilon}{\lambda_1}>0$, note that $p+1>2$, thus there
exists a constant $c_1>0$ independent of $w$ such that
$\|u_w\|\geq c_1$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
First take $u_0\in H_0^{1}(\Omega)$, by Lemma \ref{l1}
 we can construct a sequence $\{u_{n}\}$ such that
for $n\geq1$ $u_{n}$ is a nontrivial solution of the equation
\begin{equation}\label{cccc}
\begin{gathered}
-\Delta u_{n}=f(x,u_{n},\nabla u_{n-1}) , \quad \text{in }\Omega,\\
u_{n}=0, \quad \text{on }\partial\Omega.
\end{gathered}
\end{equation}
 From \eqref{cccc},  $u_{n+1}$ and $u_{n}$ satisfy
\begin{gather}\label{e3}
\int_{\Omega}\nabla u_{n+1}(\nabla u_{n+1}-\nabla
u_{n})=\int_{\Omega}f(x,u_{n+1},\nabla u_{n})(u_{n+1}-u_{n}) ,
\\
\label{e4}
\int_{\Omega}\nabla u_{n}(\nabla u_{n+1}-\nabla
u_{n})=\int_{\Omega}f(x,u_{n},\nabla u_{n-1})(u_{n+1}-u_{n}) ,
\end{gather}
 By \eqref{e3}, \eqref{e4}, (H4), Sobolev inequality, and H\"older inequality, we
 obtain
\begin{align*}
&\|u_{n+1}-u_{n}\|^{2}\\
&=\int_{\Omega}(f(x,u_{n+1},\nabla u_{n})-f(x,u_{n},\nabla u_{n-1}))(u_{n+1}-u_{n})\,dx \\
&\leq \int_{\Omega}L(|u_{n+1}-u_{n}|+|\nabla u_{n}-\nabla u_{n-1}|)|u_{n+1}-u_{n}|\,dx\\
&\leq \int_{\Omega}\frac{L}{\lambda_1}|\nabla
(u_{n+1}-u_{n})|^{2}\,dx +L(\int_{\Omega}|\nabla (u_{n}-
u_{n-1})|^{2}\,dx)^{1/2}
(\int_{\Omega}|u_{n+1}-u_{n}|^{2}\,dx)^{1/2} \\
&\leq \frac{L}{\lambda_1}\|u_{n+1}-u_{n}\|^{2}+\frac{L}{\sqrt{\lambda_1}}\|u_{n}-
u_{n-1}\|\|u_{n+1}-u_{n}\|;
\end{align*}
thus 
 $$
\|u_{n+1}-u_{n}\|\leq \frac{L\sqrt{\lambda_1}}{\lambda_1-L}\|u_{n}- u_{n-1}\|.
$$ 
Since $0<\frac{L\sqrt{\lambda_1}}{\lambda_1-L}<1$, $\{u_{n}\}$ is a
Cauchy sequence in $H_0^{1}(\Omega)$, so $\{u_{n}\}$ converges
strongly to some  $u^{\ast}\in H_0^{1}(\Omega)$.

We claim that $u^{\ast}$ is a weak solution of \eqref{cc}.
Indeed for any $\phi\in C_0^{\infty}(\Omega)$, by (H4),
\begin{align*}
&\int_{\Omega}(f(x,u_{n},\nabla u_{n-1})-f(x,u^{\ast},\nabla u^{\ast}))\phi \,dx\\
&\leq L\|\phi\|_{L^{\infty}(\Omega)}\int_{\Omega}(|u_{n}-u^{\ast}|
 +|\nabla u_{n-1}-\nabla u^{\ast}|)\,dx\\
&\leq C_{\phi}(\|u_{n}-u^{\ast}\|+\|u_{n-1}-u^{\ast}\|)
\to 0
\end{align*}
as $n\to\infty$; thus by Lemma \ref{l1},
\begin{align*}
0&=\lim_{n\to\infty}\int_{\Omega}\nabla u_{n}\nabla\phi
 -f(x,u_{n},\nabla u_{n-1})\phi \,dx\\
&=\int_{\Omega}\nabla u^{\ast}\nabla\phi-f(x,u^{\ast},\nabla
u^{\ast})\phi \,dx.
\end{align*}
Hence the claim is proved.
By Lemma \ref{l4}, $\|u_{n}\|\geq c_1$;
 thus $\|u^{\ast}\|\geq c_1$. Therefore, $u^{\ast}$ is a
nontrivial weak solution of the problem \eqref{cc}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
By Lemma \ref{l3} and using the argument as used in the proof of
Theorem \ref{thm1.1} we  complete the proof.
\end{proof}

\section{Examples}

Consider the  equation
\begin{equation}\label{c}
\begin{gathered}
-\Delta u=mu-\frac{au}{1+u^{2}}-\frac{u^{3}}{1+u^{4}}\sin^{2}|\nabla u|, 
 \quad \text{in }\Omega,\\
u=0, \quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^N(N\geq3)$ is a bounded domain with
smooth boundary. Suppose that $a>0$, $m-a<\lambda_{k}\leq m$ for
some $k\geq 1$, $m-a\neq \lambda_{j}$ for any $j\in \mathbb{N}$, and
$0<\frac{(m+a+3)\sqrt{\lambda_1}}{\lambda_1-(m+a+3)}<1$. We will
show that \eqref{c} has at least one nontrivial weak solution.
Let
$$
f(t,\xi)=mt-\frac{at}{1+t^{2}}-\frac{t^{3}}{1+t^{4}}\sin^{2}|\xi|.
$$
Then $f\in C(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$, so (H0)
holds.
Let
$$
g_0(t,\xi)=at(1-\frac{1}{1+t^{2}})-\frac{t^{3}}{1+t^{4}}\sin^{2}|\xi|.
$$
Then $f(t,\xi)=(m-a)t+g_0(t,\xi)$. It is not difficult to see that
$g_0(t,\xi)=o(|t|)$ as $t\to0$. 
Since $m-a\neq \lambda_{j}$ for any $j\in \mathbb{N}$, then (H1) holds.
Let
$$
g(t,\xi)=-\frac{at}{1+t^{2}}-\frac{t^{3}}{1+t^{4}}\sin^{2}|\xi|.
$$
Then $f(t,\xi)=mt+g(t,\xi)$. It is not difficult to see that
$g(t,\xi)=o(|t|)$ as $|t|\to\infty$. Note that
$m-a<\lambda_{k}\leq m$ for some $k\geq 1$.
 If $m\neq\lambda_{k+l}$ for any $l\geq0$, then
(H2) holds. If  $m=\lambda_{k+l}$ for some $l\geq0$, then (H3)
holds, in fact, since $a>0$, we have
$$
|g(t,\xi)|\leq|\frac{at}{1+t^{2}}|+|\frac{t^{3}}{1+t^{4}}\sin^{2}|\xi||
\leq\frac{a}{2}+1
$$
and
\begin{align*}
G(t,\xi)
&=-\int_0^{t}\frac{as}{1+s^{2}}ds-\int_0^{t}\frac{s^{3}}{1+s^{4}}\sin^{2}|\xi|ds\\
&=-\frac{a}{2}\ln(1+t^{2})-\frac{1}{4}\ln(1+t^{4})\sin^{2}|\xi|
\to-\infty
\end{align*}
as $|t|\to\infty$, uniformly in $\xi\in\mathbb{R}^{n}$.

Finally, we show that (H4) holds. By Lagrange mean value theorem,
for any $ t_1,t_2\in \mathbb{R},\xi_1, \xi_2\in
\mathbb{R}^N$ we have
\begin{align*}
&|f(t_2,\xi_2)-f(t_1,\xi_1)|\\
&=|(mt_2-\frac{at_2}{1+t_2^{2}}-\frac{t_2^{3}}{1+t_2^{4}}\sin^{2}|\xi_2|)
-(mt_1-\frac{at_1}{1+t_1^{2}}-\frac{t_1^{3}}{1+t_1^{4}}\sin^{2}|\xi_1|)|\\
&\leq m|t_2-t_1|+\big|\frac{at_2}{1+t_2^{2}}-\frac{at_1}{1+t_1^{2}}|
+|(\frac{t_2^{3}}{1+t_2^{4}}-\frac{t_1^{3}}{1+t_1^{4}})\sin^{2}|\xi_2|\big|
\\
&\quad +|\frac{t_1^{3}}{1+t_1^{4}}(\sin^{2}|\xi_2|-\sin^{2}|\xi_1|)|\\
&\leq m|t_2-t_1|+a|t_2-t_1|+3|t_2-t_1|+2|\xi_2-\xi_1|\\
&\leq (m+a+3)(|t_2-t_1|+|\xi_2-\xi_1|),
\end{align*}
so (H4) holds.
By Theorem \ref{thm1.1} and Theorem \ref{thm1.2}, Equation \eqref{c} has at
 least one nontrivial weak solution.

\subsection*{Acknowledgments}
This research is supported by the following grants:
11071098 from NSFC,
2012CB821200 from National 973 project of China,
20060183017 from SRFDP.
Also supported by the Program for New Century Excellent Talents
in University, 985 project of Jilin University, the outstanding young's
project of Jilin University,  and  Graduate Innovation Fund of Jilin
University (20111036). 

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