In this paper, we discuss multiplicity of solutions of the following boundary value problem

$\left\{ \begin{array}{ll} -\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u),&\hbox{$x\in\Omega$,} \\ u=0,&\hbox{ $x\in\partial\Omega$.} \end{array} \right.$

(1.1)

and

$\left\{ \begin{array}{ll} -\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u)+\epsilon g(x,u),&\hbox{$x\in\Omega$,} \\ u=0,&\hbox{ $x\in\partial\Omega$.} \end{array} \right.$

(1.2)

where $\Omega\subset R^N$ is a bounded domain with smooth boundary $\partial\Omega$. The function $a$ is such that $p:R\to R$ defined by

$p(t)=\left\{ \begin{array}{ll} a(|t|)t,&\hbox{$t\not=0$,} \\ 0,&\hbox{$t=0$,} \end{array} \right.$

which is an increasing homeomorphism from $R$ onto itself and the continuous function $f(x, t)\in C(\overline{\Omega}\times R, R)$ satisfies $f(x, 0)=0$, $x\in \overline{\Omega}$ and odd in $t$. $g\in C(\overline{\Omega}\times R, R)$ and $\epsilon$ is a parameter.

As we all know, when $a(t)=|t|^{p-2}$, problem (1.1) is the well known p-Laplacian equation. There is a large number of papers on the existence of solutions for p-Laplacian equation. But Problem (1.1) possesses more complicated nonlinearities, for example, it is inhomogeneous, and has important physical background, e.g.,

(a) nonlinear elasticity: $P(t)=(1+t^2)^{\gamma}-1$, $\gamma>\frac{1}{2}$.

(b) plasticity: $P(t)=t^{\alpha}(log(1+t))^{\beta}$, $\alpha\geq 1$, $\beta>0$.

(c) generalized Newtonian fluids: $P(t)=\int_0^ts^{1-\alpha}(sinh^{-1}s)^{\beta}ds$, $0\leq \alpha\leq 1$, $\beta>0$.

So in the discussions, some special techniques will be needed, and the problem (1.1) is studied in an Orlicz-Sobolev space and received considerable attention in recent years, see, for instance, the papers [1-9]. Motivated by their results, the aim of this paper is to state some multiplicity results under more general assumptions for the nonlinearity $f(x, t)$ in the symmetric case and the case that the breaking of symmetry $g(x, t)$.

The paper is organized as follows: in Section 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and give the main results. In section 3, we will give the proof for symmetric case (1.1) by symmetric mountain pass theorem, and in Section 4, with the method used in [13], we give the proof for perturbed problem (1.2).

Obviously, our problems allow a nonhomogeneous function $p$ in the differential operator. To deal with this situation, we introduce an Orlicz-Sobolev space setting as follows:

Let

$P(t)=\int_0^t p(s)ds,\hspace{2mm}\widetilde{P}(t)=\int_0^t p^{-1}(s)ds,t\in R,$

then $P$ and $\widetilde{P}$ are complementary $N$-functions (see[10]), which define the Orlicz spaces $L^P:=L^P(\Omega)$ and $L^{\widetilde{P}}:=L^{\widetilde{P}}(\Omega)$ respectively.

Throughout this paper, we assume the following condition on $P$:

$(p) 1<p^-:=\inf\limits_{t>0}\frac{tp(t)}{P(t)}\leq p^+:=\sup\limits_{t>0}\frac{tp(t)}{P(t)}<+\infty.$

Under the condition $(p)$, the Orlicz space $L^P$ coincides with the set (equivalence classes) of measurable functions $u:\Omega\to R$ such that

$\int_{\Omega}P(|u|)dx<+\infty,$

and is equipped with the (luxemburg) norm, i.e.

$|u|_P:=\inf\{k>0:\int_{\Omega}P(\frac{|u|}{k})dx<1\}.$

We will denote by $W^{1, P}(\Omega)$ the corresponding Orlicz-Sobolev space with the norm

$||u||_{W^{1, P}(\Omega)}:=|u|_P+|\nabla u|_P$

and define $W_0^{1, P}(\Omega)$ as the closure of $C_0^{\infty}$ in $W^{1, P}(\Omega)$. In this paper, we will use the following equivalent norm on $W_0^{1, P}(\Omega)$:

$||u||:=\inf\{k>0:\int_{\Omega}P(\frac{|\nabla u|}{k})dx<1\}.$

Now we introduce the Orlicz-Sobolev conjugate $P_*$ of $P$, which is given by

$P_*^{-1}(t):=\int_0^t\frac{p^{-1}(\tau)}{\tau^{\frac{N+1}{N}}}d\tau,$

where we suppose that

$\lim\limits_{t\to0}\int_t^1\frac{p^{-1}(\tau)}{\tau^{\frac{N+1}{N}}}d\tau<+\infty, \lim\limits_{t\to\infty}\int_1^t\frac{p^{-1}(\tau)}{\tau^{\frac{N+1}{N}}}d\tau=+\infty.$

Let $p_*^-:=\inf\limits_{t>0}\frac{tP_*'(t)}{P_*(t)}, p_*^+:=\sup\limits_{t>0}\frac{tP_*'(t)}{P_*(t)}.$ Throughout this paper, we assume that $p^+<p_*^-$. Now we will make the following assumptions on $f(x, t)$.

$(f_*)$ There exists an odd increasing homeomorphism $h$ from $R$ to $R$, and nonnegative constants $c_1, c_2$ such that

$|f(x, t)|\leq c_1+c_2h(|t|), \forall t\in R, \forall x\in \overline{\Omega},$

$\lim\limits_{t\to+\infty}\frac{H(t)}{P_*(kt)}=0, \forall k>0,$

(2.1)

where

$H(t):=\int_0^th(s)ds.$

Let

$\widetilde{H}(t):=\int_0^th^{-1}(s)ds,$

then we can obtain complementary N-functions which define corresponding Orlicz spaces $L^H$ and $L^{H_*}$.

Similar to condition $(p)$, we also assume the following condition on $H$:

$(h) 1<h^-:=\inf\limits_{t>0}\frac{th(t)}{H(t)}\leq h^+:=\sup\limits_{t>0}\frac{th(t)}{H(t)}<+\infty.$

Lemma 2.1 [10] Under the condition $(p)$, the spaces $L^P(\Omega)$, $W_0^{1, P}(\Omega)$ and $W^{1, P}(\Omega)$ are separable and reflexive Banach spaces.

Lemma 2.2 [10] Under the condition $(f_*)$ and (2.1), the imbedding $W_0^{1, P}(\Omega)\hookrightarrow L^H(\Omega)$ is compact.

Next we assume

$N<p_0<\liminf\limits_{t\to\infty}\frac{\log(P(t))}{\log(t)}.$

(2.2)

Lemma 2.3 Under the condition (2.2), the imbedding $W_0^{1, P}(\Omega)\hookrightarrow C(\overline{\Omega})$ is compact.}

Proof Using Lemma D.2 in [14], it follows that $W_0^{1, P}(\Omega)$ is continuously embedded in $W^{1, p^-}(\Omega)$. On the other hand, since we assume $p^->N$, we deduce that $W_0^{1, p^-}(\Omega)$ is compactly embedded in $C(\overline{\Omega})$. Thus, we obtain that $W_0^{1, P}(\Omega)$ is compactly embedded in $C(\overline{\Omega})$.

Lemma 2.4 [2] Let $\rho(u)=\int_{\Omega}P(u)dx$, we have

(1) if $|u|_P<1$, then $|u|_P^{p^+}\leq \rho(u)\leq |u|_P^{p^-}$;

(2) if $|u|_P>1$, then $|u|_P^{p^-}\leq \rho(u)\leq |u|_P^{p^+}$;

(3) if $0<t<1$, then $t^{p^+}P(u)\leq P(tu)\leq t^{p^-}P(u), $;

(4) if $t>1$, then $t^{p^-}P(u)\leq P(tu)\leq t^{p^+}P(u)$.

With all the lemmas mentioned above, we now state our results and make the following assumptions:

$(f_1)$ there exist $\eta>p^+$ and $1<\sigma<p^-$, and $a_1$, $a_2>0$, such that

$\frac{1}{\eta}f(x, t)t-F(x, t)\geq -a_1-a_2|t|^{\sigma}$

for every $t\in R$, a.e. in $\Omega$.

$(f_2)$ $\liminf\limits_{|t|\to\infty} F(x, t)/|t|^{p^+}=\infty$ uniformly a.e. in $\Omega$.

Theorem 2.1 (Symmetric case) Assume that $f(x, t)$ is odd in $t$, satisfies $(f_*)$ $(f_1)$ $(f_2)$ and $(2.1)$ with $p^-<h^+\leq p^+$, Then problem (1.1) possesses infinitely many nontrivial solutions.

Theorem 2.2 (Non-symmetric case) Assume that $f(x, t)$ is odd in $t$, satisfies $(f_*)$, $(f_1)$, $(f_2)$, $(2.1)$, $(2.2)$ with $p^-<h^+\leq p^+$, $g\in C(\overline{\Omega}\times R)$, then for any $m\in N$, there exists $\epsilon_m>0$ such that if $|\epsilon|\leq\epsilon_m$, the problem (1.2) possesses at least $m$ distinct solutions.

In this section, we assume that $N\geq 1$ and $E=W_0^{1, P}(\Omega)$, $u\in E$ is called a weak solution of problem (1.1) if

$\int_{\Omega}a(|\nabla u|)\nabla u\nabla \phi dx=\int_{\Omega}f(x, u)\phi dx, \forall \phi \in E.$

Set

$I(u)=\int_{\Omega}P(|\nabla u|)dx-\int_{\Omega}F(x, u)dx, \forall u\in E$

and we know that the critical points of $I$ are just the weak solutions of problem (1.1).

Now we state the symmetric mountain pass theorem used in this section.

Lemma 3.1 [11] Let $E$ be an infinite dimensional Banach space and let $I\in C^1(E, R)$ be even, satisfy (PS), and $I(0)=0$, if $E=V\oplus X$, where $V$ is finite dimensional, and $I$ satisfies

$(I_1)$ there are constants $\rho, \alpha>0$ such that $I|_{\partial B_{\rho}\cap X}\geq \alpha$;

$(I_2)$ for each finite dimensional subspace $\widetilde{E}\subset E$, there is an $R=R(\widetilde{E})$ such that $I\leq 0$ on $\widetilde{E}\setminus B_{R(\widetilde{E})}$, then $I$ possesses an unbounded sequence of critical values.

For E a separable and reflexive Banach space, then there exist (see [9]) $\{e_n\}_{n=1}^{\infty}\subset E$ and $\{e_n^*\}_{n=1}^{\infty}\subset E^*$ such that

$e_n^*(e_m)=\delta_{n,m}=\left\{ \begin{array}{ll} 1,&\hbox{ if $n=m$;} \\ 0,&\hbox{if $n\not=m$.} \end{array} \right. \hbox{and}\; e_n^*(v)=\alpha_n \;\hbox{for}\hspace{2mm} v=\sum\limits_{i=1}^{\infty}\alpha_i e_i\in E.$

Now we set

$V_j=\{u\in W_0^{1, P}(\Omega):e_i^*(u)=0, i>j\},\\ X_j=\{u\in W_0^{1, P}(\Omega):e_i^*(u)=0, i\leq j\},$

so

$W_0^{1, P}(\Omega)=V_j\oplus X_j.$

(3.1)

Lemma 3.2 Given $\delta>0$, there is $j\in N$ such that for all $u\in X_j$, $|u|_H\leq \delta ||u||$.

Proof We prove the lemma by contradiction. Suppose that there exist $\delta>0$ and $u_j\in X_j$ for every $j\in N$ such that $|u_j|_H\geq \delta ||u_j||$. Taking $v_j=\frac{u_j}{|u_j|_H}$, we have $|v_j|_H=1$, for every $j\in N$ and $||v_j||\leq \frac{1}{\delta}$. Hence $\{v_j\}\subset W_0^{1, P}(\Omega)$ is a bounded sequence, and we may suppose, without loss of generality, that $v_j\rightharpoonup v$ in $W_0^{1, P}(\Omega)$. Furthermore, $e_n^*(v)=0$ for every $n\in N$ since $e_n^*(v_j)=0$ for all $j\geq n$. This shows that $v=0$. On the other hand, by the compactness of embedding $W_0^{1, P}(\Omega)\hookrightarrow L^H(\Omega)$, we conclude that $|v|_H=1$. This proves the lemma.

Lemma 3.3 Suppose $f$ satisfy $(f_*)$, then there exist $j\in N$ and $\rho, \alpha >0$, such that $I|_{\partial B_{\rho}\cap X_j}\geq \alpha$.

Proof: Now suppose that $||u||>1$, from $(f_*)$, we know that

$\begin{eqnarray*} I(u)&=&\int_{\Omega}P(|\nabla u|)dx-\int_{\Omega} F(x,u)dx\\ &\geq& ||u||^{p^-}-C_1|u|_H^{h^+}-C_2. \end{eqnarray*}$

Consequently, considering $\delta>0$ to be chosen posteriorly by Lemma 3.2, we have for all $u\in X_j$ and $j$ sufficiently large,

$I(u)\geq ||u||^{p^-}(1-C_1\delta^{h^+}||u||^{h^+-p^-})-C_2.$

Now, taking $||u||=\rho(\delta)=(\frac{1}{2C\delta^{h^+}})^{\frac{1}{h^+-p^-}}$, and noting that $\rho(\delta)\to+\infty$, if $\delta\to0$. We can choose $\delta>0$ such that $\frac{1}{2}\rho^{p^-}>C_2$, $\rho>1$, and $I(u)>0$ for every $u\in X_j$, $||u||=\rho$, the proof is complete.

Lemma 3.4 Suppose $f$ satisfy $(f_2)$, then for each finite dimensional subspace $\widetilde{E}\subset E$, there is an $R=R(\widetilde{E})$ such that $I\leq 0$ on $\widetilde{E}\setminus B_{R(\widetilde{E})}$.

Proof: From condition $(f_2)$, given $L>0$, there is a $C>0$ such that for every $t\in R$, a.e. $x\in\Omega$,

$F(x, t)\geq L|t|^{p^+}-C.$

Now let $\widetilde{E}$ be a finite dimensional subspace, suppose that $u\in \widetilde{E}$ with $||u||>1$, then

$\begin{eqnarray*} I(u)&=&\int_{\Omega}P(\nabla u|)dx-\int_{\Omega} F(x, u)dx&\leq& ||u||^{p^+}-L\int_{\Omega}|u|^{p^+}dx +C&\leq& (1-CL)||u||^{p^+}+C. \end{eqnarray*}$

Choose $L$ large enough such that $1-LC<0$ then there exists $R(\widetilde{E})>1$ such that $I(u)\leq0$ for all $||u||\geq R(\widetilde{E})$ and the proof is complete.

Lemma 3.5 Suppose $f$ satisfies $(f_1)$, then $I$ satisfies (PS) condition.}

Proof We suppose that $||u_n||>1$,

$\begin{aligned} M+o(1)||u_n|| & \geq I(u_n)-\frac{1}{\eta}I'(u_n)u_n \\ & = \int_{\Omega}P(|\nabla u_n|)dx-\frac{1}{\eta}\int_{\Omega}p(|\nabla u_n|)\nabla u_n dx+\int_{\Omega}(\frac{1}{\eta}f(x, u_n)u_n-F(x, u_n)) dx \\ & \geq (1-\frac{p^+}{\eta})||u_n||^{p^-}-a_1|\Omega|-C||u_n||^{\sigma}, \end{aligned}$

Noting that $1<\sigma <p^-$, $\eta>p^+$, $\{u_n\}$ is bounded. By [9] Lemma 3.1, we know that $I$ satisfies the (PS) condition.

Proof of Theorem 1.1 First, we recall that $W_0^{1, P}(\Omega)=V_j\oplus X_j$, where $V_j$ and $X_j$ are defined in (3.1). Invoking Lemma 3.3, we find $j\in N$, and $I$ satisfies $(I_1)$ with $X=X_j$. Now by Lemma 3.4 $I$ satisfies $(I_2)$. Since $I(0)=0$ and $I$ is even, we may apply Lemma 3.1 to conclude that $I$ possesses infinitely many nontrivial critical points. The proof is complete.

First of all, let us recall some notions and facts from Degiovanni and Lancelotti [12]. Let $E$ be a Banach space and $I\in C^1(E, R)$. For $b\in \widetilde{R}:=R\cup\{-\infty, +\infty\}$, set $I^b=\{u\in E|I(u)\leq b\}$.

Definition 4.1 [12] Let $a$, $b\in\widetilde{R}$ with $a\leq b$. The pair $(I^b, I^a)$ is said to be trivial, if for every neighborhood $[\alpha', \alpha'']$ of $a$ and $[\beta', \beta'']$ of $b$ $(\alpha', \alpha'', \beta', \beta''\in R)$ there exist two closed subsets $A$ and $B$ such that $I^{\alpha'}\subset A\subset I^{\alpha''}$, $I^{\beta'}\subset B\subset I^{\beta''}$ and $A$ is a strong deformation retract of $B$.

Definition 4.2 [12] A real number $c$ is said to be an essential value of $I$, if for every $\epsilon>0$, there exist $a, b\in(c-\epsilon, c+\epsilon)$ with $a<b$ such that the pair $(I^b, I^a)$ is not trivial.

Lemma 4.1 [12] Let $a$, $b\in \widetilde{R}$ with $a<b$. Let us assume that $I$ has no essential value in $(a, b)$. Then the pair $(I^b, I^a)$ is trivial.

Lemma 4.2 [12] Let $c$ be an essential value of $I$. Then for every $\epsilon>0$ there exists $\delta>0$ such that every $J\in C^1(E, R)$ with $sup\{|J(u)-I(u)||u\in E\}<\delta$ admits an essential value in $(c-\epsilon, c+\epsilon)$.

Lemma 4.3 [12] Let $c$ be an essential value of $I$. If $(PS)_c$ holds for $I$, then $c$ is a critical value of $I$.

Proof of Theorem 1.2 Fix a number $m\in N$. For $k\in N$, choose a continuous function $\beta_k(t)=1$ if $|t|\leq k$, $\beta_k(t)=0$ if $|t|\geq k+1$ and $0<\beta_k(t)<1$, if $k<|t|<k+1$. Let $g_k(x, t)=\beta_k(t)g(x, t)$ and $G_k(x, t)=\int_0^t g_k(x, s)ds$. For any $k\in N$, choose $\epsilon_1(k)>0$ such that for all $x\in \Omega$ and $t\in R$,

$\epsilon_1(k)|g_k(x, t)|<1, \epsilon_1(k)|G_k(x, t)|<1, \epsilon_1(k)tg_k(x, t)<1.$

For $k\in N$ and $|\epsilon|\leq\epsilon_1(k)$, set

$I(u)=\int_{\Omega}P(|\nabla u|)dx-\int_{\Omega} F(x, u)dx, u\in W_0^{1, P}(\Omega),$

and

$I_{\epsilon k}(u)=\int_{\Omega}P(|\nabla u|)dx-\int_{\Omega}(F(x, u)+\epsilon G_k(x, u))dx, u\in W_0^{1, P}(\Omega).$

Then $I$ and $I_{\epsilon k}$ satisfies $(PS)_c$ for every real number $c$. Denote $E_k=span\{e_1, e_2, \cdots e_k\}$. By $(f_2)$, there exists an increasing sequence of positive numbers $\{R_k\}$ such that

$I(u)\leq 0, \forall u\in E_k, ||u||\geq R_k.$

Let $D_k=\{u|u\in E_k, ||u||\leq R_k\}$, and $\partial D_k$ be the boundary of $D_k$ in $E_k$.

Define a sequence $\{\Phi_k\}$ of sets of functions inductively as

$\Phi_1=\{h|h\in C(D_1, E), \hbox{h is odd, and }h|_{\partial D_1}=id\},$

and for $k=1, 2, \cdots$,

$\Phi_{k+1}=\{h|h\in C(D_{k+1}, E), \hbox{h is odd, } h|_{\partial D_{k+1}}=id, \hbox{and }h|_{D_k}\in \Phi_k\}.$

Define for $k=1, 2\cdots$

$b_k=\inf\limits_{h\in \Phi_k}\max\limits_{u\in D_k} I(h(u)).$

It is obvious that $b_1\leq b_2\leq b_3\leq \cdots$.Define for $k=1, 2\cdots$,

$\begin{gathered} {G_k} = \{ h|h \in C({D_k},E),h{\text{ is odd, and }}h{|_{\partial {D_k}}} = id\} , \hfill \\ {\Gamma _k} = \{ h(\overline {{D_j} \setminus Y} )|h \in {G_j},j \geqslant k,Y \in \Sigma ,\;{\text{and}}\;\gamma (Y) \leqslant j - k\} , \hfill \\ \end{gathered} $

and

$c_k=\inf\limits_{B\in \Gamma_k}\max\limits_{u\in B} I(u).$

where $\Sigma$ is the class of closed symmetric subsets of $E$ and $\gamma(Y)$ is the genus of $Y$ for $Y\in\Sigma$. By Rabinowitz [11, Proposition 9.33], $c_k\to\infty$, as $k\to\infty$. From the definition of $b_k$ and $c_k$, it is clear that $b_k\geq c_k$ for all $k\in N$. So $b_k\to\infty$ as $k\to\infty$. Let $\Lambda =\{c\in R, c \hspace{2mm}\hbox{is an essential value of} \hspace{2mm}I\}$. Now we prove that $\Lambda\not=\emptyset$, and $\sup\Lambda=+\infty$. If this statement was false, then there would exists $k\in N$ such that $0<b_k<b_{k+1}$ and $[b_k, +\infty)\cap \Lambda=\emptyset$. Choose real numbers $\alpha'$, $a$, $\alpha''$ such that

$b_k<\alpha'<a<\alpha''<b_{k+1}.$

(4.1)

Let $h\in \Phi_k$ be such that $\max_{u\in D_k} I(h(u))<\alpha'$. for $k\in N$, define

$D_{k+1}^+=\{u|u=v+te_{k+1}, v\in E_k, t\geq0, ||u||\leq R_{k+1}\},$

and let $\partial D_{k+1}^+$ be the boundary of $D_{k+1}^+$ in $E_{k+1}$. Extend $h$ to be a function $h_1\in C(\partial D_{k+1}^+, E)$ as

$h_1(u)=\left\{ \begin{array}{ll} h(u),&\hbox{$u\in D_k$;} \\ u,&\hbox{$u\in\partial D_{k+1}^+\setminus D_k$.} \end{array} \right.$

Clearly, $h_1$ is well defined, continuous and $h_1(\partial D_{k+1}^+)\subset I^{\alpha'}$. Extend $h_1$ to a function $h_2\in C(D_{k+1}^+, E)$ and let $\beta=\max\{I(h_2(u))|u\in D_{k+1}^+\}$. By lemma 4.1, the pair $(I^{+\infty}, I^a)$ is trivial. So there exist closed subsets $A$ and $B$ of $E$ such that $I^{\alpha'}\subset A\subset I^{\alpha''}$, $I^{\beta}\subset B$, and there exists a strong deformation interaction $\eta:B\times[0,1]\to B$ of $B$ to $A$. Define $h_3\in C(D_{k+1}^+, E)$ as $h_3(u)=\eta(h_3(u), 1)$ then $h_3(u)$ satisfies

$I(h_3(D_{k+1}^+))\subset I^{\alpha''},$

(4.2)

$h_3\hbox{ is odd on }D_{k+1}^+\cap E_k,$

(4.3)

$h_3=id \hspace{2mm}\hbox{on }\partial D_{k+1}^+\cap \partial D_{k+1},$

(4.4)

$h_3|_{D_k}=h.$

(4.5)

Define $h_4\in C(D_{k+1}, E)$ as

$h_4(u)=\left\{ \begin{array}{ll} h_3(u),&\hbox{$u\in D_{k+1}^+$;} \\ -h_3(-u),&\hbox{$u\in D_{k+1}\setminus D_{k+1}^+$.} \end{array} \right.$

Then (4.3) implies that $h_4$ is odd, (4.4) implies $h_4|_{\partial D_{k+1}}=id, $ and (4.5) implies $h_4|_{D_k}\in\Phi_k$. So $h_4\in \Phi_{k+1}$ which is a contradiction, since by (4.1) and (4.2) we have

$b_{k+1}\leq\max\limits_{u\in D_{k+1}}I(h_4(u))=\max\limits_{u\in D_{k+1}^+}I(h_3(u))\leq\alpha ''<b_{k+1}.$

Therefore $\Lambda\not=\emptyset$ and $\sup \Lambda=+\infty$. Choose a strictly increasing sequence of positive number $\{d_k\}\subset \Lambda$, such that $d_k\to+\infty$ as $k\to+\infty$. By Lemma 4.2 and the definition of $I_{\epsilon k}$, there exists $\epsilon_2(k)\in(0, \epsilon_1(k))$ such that if $|\epsilon|\leq\epsilon_2(k)$, then $I_{\epsilon k}$ has at least $m$ essential values $d_{\epsilon k1}$, $d_{\epsilon k2}$, $\cdots$, $d_{\epsilon km}$ such that

$0<d_{\epsilon k1}<d_{\epsilon k2}<\cdots<d_{\epsilon km}<d_{m+1}.$

According to Lemma 4.3, there are $m$ distinct critical points $u_{\epsilon ki}$ $(i=1, 2, \cdots, m)$ of $I_{\epsilon k}$ such that

$\int_{\Omega}P(|\nabla u_{\epsilon ki}|)dx-\int_{\Omega}(F(x, u_{\epsilon ki})+\epsilon G_k(x, u_{\epsilon ki}))dx=d_{\epsilon ki},$

and

$\int_{\Omega}p(|\nabla u_{\epsilon ki}|)\nabla u_{\epsilon ki}dx=\int_{\Omega}(f(x, u_{\epsilon ki})u_{\epsilon ki}+\epsilon g_k(x, u_{\epsilon ki})u_{\epsilon ki})dx.$

Then there exists a constant $C_m>0$ independent of $\epsilon$ and $k$ such that for every $k\in N$, $|\epsilon|\leq \epsilon_2(k)$ and $i=1, 2, \cdots, m$,

$\int_{\Omega}P(|\nabla u_{\epsilon ki}|)dx-\int_{\Omega}(F(x, u_{\epsilon ki})+\epsilon G_k(x, u_{\epsilon ki}))dx\leq C_m,\\ \int_{\Omega}p(|\nabla u_{\epsilon ki}|)\nabla u_{\epsilon ki}dx-\int_{\Omega}f(x, u_{\epsilon ki})u_{\epsilon ki}\geq-C_m.$

Then we have for every $k\in N$, $|\epsilon|\leq \epsilon_2(k)$, and $i=1, 2, \cdots m$,

$||u_{\epsilon ki}||\leq C_m'.$

(4.6)

where $C_m'$ is a constant independent of $\epsilon$ and $k$ and so there exists a constant $C_m''$ independent of $\epsilon$ and $k$ such that for every $k\in N$, $|\epsilon|\leq \epsilon_2(k)$ and $i=1, 2\cdots, m$,

$||u_{\epsilon ki}||_{C(\overline{\Omega})}\leq C_m''.$

(4.7)

So if $k>C_m''$ then for any $\epsilon$ with $|\epsilon|\leq \epsilon_2(k)$ the problem possesses $m$ distinct solutions $u_{\epsilon k1}$, $u_{\epsilon k2}$, $\cdots$, $u_{\epsilon km}$. The proof is complete.

Remark 4.1 In this theorem we assume (2.2) just because we can get (4.7) from (4.6) by the embedding result such that $C_m''$ is independent of $\epsilon$ and $k$.