Multiplicity of solutions for Kirchhoff-type problems involving critical growth

Abstract

In this paper, by using the concentration-compactness principle and the variational method, we obtain a multiplicity result for Kirchhoff-type problems involving critical growth in bounded domains.

MSC: 35J70, 35B20.

1 Introduction

In this paper we deal with the existence and multiplicity of solutions to the following Kirchhoff-type problems involving the critical growth:

$$\{\begin{array}{ll}-(a+b{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}dx)\mathrm{\Delta }u-a[\mathrm{\Delta }(u^2)]u=u^{2(2^{\ast })-1}+\lambda h(x,u),& x\in \mathrm{\Omega },\\ u=0,& x\in \partial \mathrm{\Omega },\end{array}$$

(1.1)

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) is an open bounded domain with smooth boundary and λ is a positive parameter. The number $2^{\ast }=2N/(N-2)$ is the critical exponent according to the Sobolev embedding.

Much interest has arisen in problems involving critical exponents, starting from the celebrated paper by Brezis and Nirenberg [1]. For example, Li and Zou [2] obtained infinitely many solutions with odd nonlinearity. Chen and Li [3] obtained the existence of infinitely many solutions by using minimax procedure. For more related results, we refer the interested readers to [4]–[10] and references therein.

On the one hand, without $a[\mathrm{\Delta }(u^2)]$, (1.1) reduces to the following Dirichlet problem of Kirchhoff type:

$$\{\begin{array}{ll}-(a+b{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}dx)\mathrm{\Delta }u=f(x,u),& x\in \mathrm{\Omega },\\ u{|}_{\partial \mathrm{\Omega }}=0,\end{array}$$

(1.2)

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$, problem (1.2) is a generalization of a model introduced by Kirchhoff [11]. More precisely, Kirchhoff proposed a model given by the equation

$$\rho \frac{{\partial }^{2}u}{\partial t^2}-(\frac{{\rho }_0}{h}+\frac{E}{2L}{\int }_0^L|\frac{\partial u}{\partial x}{|}^{2}dx)\frac{{\partial }^{2}u}{\partial x^2}=0,$$

(1.3)

where ρ, ${\rho }_0$, h, E, L are constants, which extends the classical D’Alembert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. Equation (1.2) is related to the stationary analog of problem (1.3). Problem (1.2) received much attention only after Lions [12] proposed an abstract framework to study the problem. Some important and interesting results can be found; see for example [13]–[17]. We note that results dealing with problem (1.2) with critical nonlinearity are relatively scarce [18]–[21].

In [22], by means of a direct variational method, the authors proved the existence and multiplicity of solutions to a class of p-Kirchhoff-type problem with Dirichlet boundary data. In [23], the authors showed the existence of infinite solutions to the p-Kirchhoff-type quasilinear elliptic equation. But they did not give any further information on the sequence of solutions. Recently, Kajikiya [24] established a critical point theorem related to the symmetric mountain-pass lemma and applied to a sublinear elliptic equation. However, there are no such results on Kirchhoff-type problems (1.1).

On the other hand, there are many papers concerned with the following quasilinear elliptic equations:

$$-\mathrm{\Delta }u+V(x)u-\left[\mathrm{\Delta }\left(u^2\right)\right]u=h(x,u),x\in {\mathbb{R}}^N.$$

(1.4)

Such equations arise in various branches of mathematical physics and they have been the subject of extensive study in recent years. In [25], by a change of variables the quasilinear problem was transformed to a semilinear one and an Orlicz space framework was used as the working space, and they were able to prove the existence of positive solutions of (1.4) by the mountain-pass theorem. The same method of a change of variables was used in [26], but the usual Sobolev space $H^1(\mathrm{\Omega })$ framework was used as the working space and one studied a different class of nonlinearity. In [27], the existence of both one sign and nodal ground state-type solutions was established by the Nehari method.

Motivated by the reasons above, the aim of this paper is to show the existence of infinitely many soliton solutions of problem (1.1), and there exists a sequence of infinitely many arbitrarily small soliton solutions converging to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya [24].

Note that $2(2^{\ast })$ behaves like a critical exponent for the above equations; see [25]. For the subcritical case, the existence of solutions for problem (1.4) was studied in [25]–[28] and it was left open for the critical exponent case; see [25]. To the best of our knowledge, the existence of non-trivial radial solutions for (1.4) with $h(u)=\mu u^{2(2^{\ast })-1}$ was firstly studied by Moameni [29], where the same Orlicz space as [25] was used. In [30], the authors showed the existence of multiple solutions for problems (1.1) with $a=1$ and $b=0$ by minimax methods and the Krasnoselski genus theory. For other interesting results see [31], [32].

To the best of our knowledge, the existence and multiplicity of soliton solutions to problem (1.1) has never been studied by variational methods. As we shall see in the present paper, problem (1.1) can be viewed as an elliptic equation coupled with a non-local term. The competing effect of the non-local term with the critical nonlinearity and the lack of compactness of the embedding of $H^1(\mathrm{\Omega })$ into the space $L^p(\mathrm{\Omega })$ prevent us from using the variational methods in a standard way. Some new estimates for such a Kirchhoff equation involving Palais-Smale sequences, which are key points in the application of this kind of theory, need to be established. We mainly follow the idea of [24], [33]. Let us point out that, although the idea was used before for other problems, the adaptation of the procedure to our problem is not trivial at all; because of the appearance of a non-local term, we must consider our problem for a suitable space and so we need more delicate estimates.

Our main result in this paper is the following.

Theorem 1.1

Suppose that$h(x,u)$satisfies the following conditions:

(${\mathrm{H}}_1^{\ast }$):$h(x,u)\in C(\mathrm{\Omega }\times \mathbb{R},\mathbb{R})$, $h(x,-u)=-h(x,u)$for all$u\in \mathbb{R}$;

(${\mathrm{H}}_2^{\ast }$):${lim}_{|u|\to \mathrm{\infty }}\frac{h(x,u)}{{|u|}^{2(2^{\ast })-1}}=0$uniformly for$x\in \mathrm{\Omega }$;

(${\mathrm{H}}_3^{\ast }$):${lim}_{|u|\to 0^{+}}\frac{h(x,u)}{u}=\mathrm{\infty }$uniformly for$x\in \mathrm{\Omega }$.

Then there exists${\lambda }^{\ast }>0$such that for any$\lambda \in (0,{\lambda }^{\ast })$, problem (1.1) has a sequence of non-trivial solutions$\{u_n\}$and$u_n\to 0$in$H_0^1(\mathrm{\Omega })$as$n\to \mathrm{\infty }$.

2 Preliminary lemmas

The energy functional corresponding to problem (1.1) is defined as follows:

$$\begin{array}{rcl}J(u)& :=& \frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}dx+\frac{b}{4}{\left({\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}dx\right)}^2+a{\int }_{\mathrm{\Omega }}{|u|}^2{|\mathrm{\nabla }u|}^{2}dx\\ -\frac{1}{2(2^{\ast })}{\int }_{\mathrm{\Omega }}{|u|}^{2(2^{\ast })}dx-\lambda {\int }_{\mathrm{\Omega }}H(x,u)dx\\ =& \frac{a}{2}{\int }_{\mathrm{\Omega }}(1+2{|u|}^2){|\mathrm{\nabla }u|}^{2}dx+\frac{b}{4}{\left({\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}dx\right)}^2-\frac{1}{2(2^{\ast })}{\int }_{\mathrm{\Omega }}{|u|}^{2(2^{\ast })}dx\\ -\lambda {\int }_{\mathrm{\Omega }}H(x,u)dx,\end{array}$$

where $H(x,s)={\int }_0^{s}h(x,\tau )d\tau$ for $(x,s)\in {\mathbb{R}}^N\times \mathbb{R}$. It should be pointed out that the functional J is not well defined in general, for instance, in $H_0^1(\mathrm{\Omega })$. To overcome this difficulty, we employ an argument developed by Colin and Jeanjean [26]. We make the change of variables $v=f^{-1}(u)$, where f is defined by

$$f^{\prime }(t)=\frac{1}{\sqrt{1+2f^2(t)}}\text{and}f(0)=0$$

on $[0,+\mathrm{\infty })$ and by $f(t)=-f(-t)$ on $(-\mathrm{\infty },0]$.

The following result is due to Colin and Jeanjean [26] (see also [34]).

Lemma 2.1

The function f satisfies the following properties:

(f₀):f is uniquely defined$C^{\mathrm{\infty }}$and invertible.

(f₁):$|f^{\prime }(t)|\le 1$for all$t\in \mathbb{R}$.

(f₂):$\frac{f(t)}{t}\to 1$as$t\to 0$.

(f₃):$\frac{f(t)}{\sqrt{t}}\to 2^{\frac{1}{4}}$as$t\to \mathrm{\infty }$.

(f₄):$\frac{1}{2}f(t)\le t{f}^{\prime }(t)\le f(t)$for all$t\ge 0$.

(f₅):$|f(t)|\le t$for all$t\in \mathbb{R}$.

(f₆):$|f(t)|\le 2^{\frac{1}{4}}{|t|}^{\frac{1}{2}}$for all$t\in \mathbb{R}$.

(f₇):The function$f^2(t)$is strictly convex.

(f₈):There exists a positive constant C such that

$$|f(t)|\ge \{\begin{array}{ll}C|t|,& |t|\le 1,\\ C{|t|}^{\frac{1}{2}},& |t|\ge 1.\end{array}$$

(f₉):There exist positive constants$C_1$and$C_2$such that

$$|t|\le C_1|f(t)|+C_2{|f(t)|}^2\mathit{\text{for all }}t\in \mathbb{R}.$$

(f₁₀):$|f(t)f^{\prime }(t)|\le \frac{1}{\sqrt{2}}$for all$t\in \mathbb{R}$.

So after this change of variables, we can write $J(u)$ as

$$\begin{array}{rcl}J(v)& :=& \frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx+\frac{b}{4}{\left({\int }_{\mathrm{\Omega }}{|f^{\prime }(v)|}^2{|\mathrm{\nabla }v|}^{2}dx\right)}^2-\frac{1}{2(2^{\ast })}{\int }_{\mathrm{\Omega }}{|f(v)|}^{2(2^{\ast })}dx\\ -\lambda {\int }_{\mathrm{\Omega }}H(x,f(v))dx.\end{array}$$

(2.1)

Then $J(v)$ is well defined on $H_0^1(\mathrm{\Omega })$. Standard arguments [35], [36] show that $J(v):H_0^1(\mathrm{\Omega })\to \mathbb{R}$ is of class $C^1$ with

$$\begin{array}{rcl}〈J^{\prime }(v),w〉& =& a{\int }_{\mathrm{\Omega }}\mathrm{\nabla }v\mathrm{\nabla }wdx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })-1}(v)f^{\prime }(v)wdx-\lambda {\int }_{\mathrm{\Omega }}h(x,f(v))f^{\prime }(v)wdx\\ +\frac{b}{2}\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }v|}^2}{1+2f^2(v)}dx\right)\left({\int }_{\mathrm{\Omega }}\frac{2\mathrm{\nabla }v\mathrm{\nabla }w(1+2f^2(v))-4{|\mathrm{\nabla }v|}^{2}f(v)f^{\prime }(v)w}{{[1+2f^2(v)]}^2}dx\right),\end{array}$$

for $v,w\in H_0^1(\mathrm{\Omega })$.

As in [26], we note that if v is a non-trivial critical point of J, then v is a non-trivial solution of the problem

$$-a\mathrm{\Delta }v-b{\int }_{{\mathbb{R}}^N}{|f^{\prime }(v)|}^2{|\mathrm{\nabla }v|}^{2}dx\cdot \eta (v)=g(x,v),$$

(2.2)

where

$$\eta (v)=(2f^{\prime }(v)f^{″}(v){|\mathrm{\nabla }v|}^2+f^{\prime 2}(v)\mathrm{\Delta }v+2f(v)f^{\prime 5}(v){|\mathrm{\nabla }v|}^2)$$

and

$$g(x,v)=f^{\prime }(v)(\lambda h(x,f(v))+f^{2(2^{\ast })-1}(v)).$$

Therefore, let $u=f(v)$ and since ${(f^{-1})}^{\prime }(t)={[f^{\prime }(f^{-1}(t))]}^{-1}=\sqrt{1+2t^2}$, we conclude that u is a non-trivial solution of the problem

$$-(a+b{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^{2}dx)\mathrm{\Delta }u-a\left[\mathrm{\Delta }\left(u^2\right)\right]u=u^{2(2^{\ast })-1}+\lambda h(x,u).$$

The auxiliary result of this paper is as follows.

Theorem 2.1

Suppose that$h(x,s)$satisfies the following conditions:

(H₁):$h(x,s)\in C(\mathrm{\Omega }\times \mathbb{R},\mathbb{R})$, $h(x,-s)=-h(x,s)$for all$s\in \mathbb{R}$;

(H₂):${lim}_{|s|\to \mathrm{\infty }}\frac{h(x,s)}{{|s|}^{2(2^{\ast })-1}}=0$uniformly for$x\in \mathrm{\Omega }$;

(H₃):${lim}_{|u|\to 0^{+}}\frac{h(x,s)}{s}=\mathrm{\infty }$uniformly for$x\in \mathrm{\Omega }$.

Then there exists${\lambda }^{\ast }>0$such that for any$\lambda \in (0,{\lambda }^{\ast })$, problem (2.2) has a sequence of non-trivial solutions$\{v_n\}$and$v_n\to 0$in$H_0^1(\mathrm{\Omega })$as$n\to \mathrm{\infty }$.

We recall the second concentration-compactness principle of Lions [37].

Lemma 2.2

Let$\{v_n\}\subset H_0^1(\mathrm{\Omega })$be a weakly convergent sequence to v in$H_0^1(\mathrm{\Omega })$such that${|v_n|}^{2^{\ast }}\rightharpoonup \nu$and${|\mathrm{\nabla }{v}_n|}^2\rightharpoonup \mu$in the sense of measures. Then, for some at most countable index set I,

• (i) $\nu ={|v|}^{2^{\ast }}+{\sum }_{j\in I}{\delta }_{x_j}{\nu }_j$, ${\nu }_j>0$,

• (ii) $\mu \ge {|\mathrm{\nabla }v|}^2+{\sum }_{j\in I}{\delta }_{x_j}{\mu }_j$, ${\mu }_j>0$,

• (iii) ${\mu }_j\ge S{\nu }_j^{2/2^{\ast }}$,

where S is the best Sobolev constant, i.e. $S=inf\{{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx:{\parallel v\parallel }_{2^{\ast }}^{2^{\ast }}=1\}$, $x_j\in {\mathbb{R}}^N$, ${\delta }_{x_j}$are Dirac measures at$x_j$and${\mu }_j$, ${\nu }_j$are constants.

Under assumptions (H₁) and (H₂), we have

$$h(x,s)s=o\left({|s|}^{2(2^{\ast })}\right),H(x,s)=o\left({|s|}^{2(2^{\ast })}\right),$$

which means that, for all $\epsilon >0$, there exist $a(\epsilon ),b(\epsilon )>0$ such that

$$|h(x,s)s|\le a(\epsilon )+\epsilon {|s|}^{2(2^{\ast })},$$

(2.3)

$$|H(x,s)|\le b(\epsilon )+\epsilon {|s|}^{2(2^{\ast })}.$$

(2.4)

Hence,

$$H(x,s)-\frac{1}{4}h(x,s)s\le c(\epsilon )+\epsilon {|s|}^{2(2^{\ast })},$$

(2.5)

for some $c(\epsilon )>0$.

Lemma 2.3

Assume conditions (H₁) and (H₂) hold. Then for any$\lambda >0$, the functional J satisfies the local (PS) _c condition in

$$c\in (-\mathrm{\infty },\frac{1}{2N}{\left(2^{-1}aS\right)}^{\frac{N}{2}}-\lambda c\left(\frac{1}{4N\lambda }\right)|\mathrm{\Omega }|)$$

in the following sense: if

$$J(v_n)\to c<\frac{1}{2N}{\left(2^{-1}aS\right)}^{\frac{N}{2}}-\lambda c\left(\frac{1}{4N\lambda }\right)|\mathrm{\Omega }|$$

and$J^{\prime }(v_n)\to 0$for some sequence in$H_0^1(\mathrm{\Omega })$, then$\{v_n\}$contains a subsequence converging strongly in$H_0^1(\mathrm{\Omega })$.

Proof

Let $\{v_n\}$ be a sequence in $H_0^1(\mathrm{\Omega })$ such that

$$\begin{array}{rl}J(v_n)=& \frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{v}_n|}^{2}dx+\frac{b}{4}{\left({\int }_{\mathrm{\Omega }}{|f^{\prime }(v_n)|}^2{|\mathrm{\nabla }{v}_n|}^{2}dx\right)}^2-\frac{1}{2(2^{\ast })}{\int }_{\mathrm{\Omega }}{|f(v_n)|}^{2(2^{\ast })}dx\\ -\lambda {\int }_{\mathrm{\Omega }}H(x,f(v_n))dx=c+o(1),\end{array}$$

(2.6)

$$\begin{array}{rl}〈J^{\prime }(v_n),w〉=& a{\int }_{\mathrm{\Omega }}\mathrm{\nabla }{v}_n\mathrm{\nabla }wdx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })-1}(v_n)f^{\prime }(v_n)wdx\\ -\lambda {\int }_{\mathrm{\Omega }}h(x,f(v_n))f^{\prime }(v_n)wdx+b\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)\\ \times \left({\int }_{\mathrm{\Omega }}\frac{2\mathrm{\nabla }{v}_n\mathrm{\nabla }w(1+2f^2(v_n))-4{|\mathrm{\nabla }{v}_n|}^{2}f(v_n)f^{\prime }(v_n)w}{{[1+2f^2(v_n)]}^2}dx\right)\\ =& o(1)\parallel v_n\parallel .\end{array}$$

(2.7)

Choose $w=w_n=\sqrt{1+2f^2(v_n)}f(v_n)$, we have $w_n\in H_0^1(\mathrm{\Omega })$ and

$$|\mathrm{\nabla }{w}_n|=(1+\frac{2f^2(v_n)}{1+2f^2(v_n)})|\mathrm{\nabla }{v}_n|.$$

Thus, we can deduce that $\parallel w_n\parallel \le c\parallel v_n\parallel$. By (2.7) we have

$$\begin{array}{rcl}〈J^{\prime }(v_n),w_n〉& =& a{\int }_{\mathrm{\Omega }}(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}){|\mathrm{\nabla }{v}_n|}^{2}dx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\\ +b{\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)}^2-\lambda {\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)dx\\ =& o(1)\parallel v_n\parallel .\end{array}$$

(2.8)

By (2.6) and (2.8), we have

$$\begin{array}{rcl}c+o(1)\parallel v_n\parallel & =& J(v_n)-\frac{1}{4}〈J^{\prime }(v_n),w_n〉\\ =& \frac{a}{4}{\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx+(\frac{1}{4}-\frac{1}{2(2^{\ast })}){\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\\ -\lambda {\int }_{\mathrm{\Omega }}H(x,f(v_n))dx+\frac{\lambda }{4}{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)dx\\ \ge & \frac{1}{2N}{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx-\lambda {\int }_{\mathrm{\Omega }}H(x,f(v_n))dx+\frac{\lambda }{4}{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)dx,\end{array}$$

i.e.

$$\frac{1}{2N}{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\le \lambda {\int }_{\mathrm{\Omega }}(H(x,f(v_n))-\frac{1}{4}h(x,f(v_n))f(v_n))dx+c+o(1)\parallel v_n\parallel .$$

Then by (2.5), we have

$$(\frac{1}{2N}-\lambda \epsilon ){\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\le \lambda c(\epsilon )|\mathrm{\Omega }|+c+o(1)\parallel v_n\parallel .$$

Setting $\epsilon =1/4N\lambda$, we get

$${\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\le M+o(1)\parallel v_n\parallel ,$$

(2.9)

where $o(1)\to 0$ and M is a some positive number. On the other hand, by (2.4) and (2.6), we have

$$\begin{array}{rcl}c+o(1)\parallel v_n\parallel & =& J(v_n)=\frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{v}_n|}^{2}dx+\frac{b}{4}{\left({\int }_{\mathrm{\Omega }}{|f^{\prime }(v_n)|}^2{|\mathrm{\nabla }{v}_n|}^{2}dx\right)}^2\\ -\frac{1}{2(2^{\ast })}{\int }_{\mathrm{\Omega }}{|f(v_n)|}^{2(2^{\ast })}dx-\lambda {\int }_{\mathrm{\Omega }}H(x,f(v_n))dx\\ \ge & \frac{a}{2}{\parallel v_n\parallel }^2-\lambda b(\epsilon )|\mathrm{\Omega }|-[\frac{1}{2(2^{\ast })}+\lambda \epsilon ]{\int }_{\mathrm{\Omega }}{|f(v_n)|}^{2(2^{\ast })}dx.\end{array}$$

(2.10)

Therefore, the inequalities (2.9) and (2.10) imply that $\{v_n\}$ is bounded in $H_0^1(\mathrm{\Omega })$. Then $\{f(v_n)\}$ is also bounded in $H_0^1(\mathrm{\Omega })$. Therefore we can assume that $v_n\rightharpoonup v$ in $H_0^1(\mathrm{\Omega })$, $v_n\to v$ a.e. in Ω, since $f\in C^{\mathrm{\infty }}$, then $f^2(v_n)\to f^2(v)$ a.e. in Ω and then $f^2(v_n)\rightharpoonup f^2(v)$ in $H_0^1(\mathrm{\Omega })$. Thus, there exist measures μ and ν such that ${|\mathrm{\nabla }{f}^2(v_n)|}^2\rightharpoonup \mu$, $f^{2(2^{\ast })}(v_n)\rightharpoonup \nu$. Let $x_j$ be a singular point of the measures μ and ν. We define a function $\varphi (x)\in C_0^{\mathrm{\infty }}({\mathbb{R}}^N)$ such that $\varphi (x)=1$ in $B(x_j,ϵ)$, $\varphi (x)=0$ in $\mathrm{\Omega }\setminus B(x_j,2ϵ)$ and $|\mathrm{\nabla }\varphi |\le 2/ϵ$ in Ω. Let ${\tilde{w}}_n=\sqrt{1+2f^2(v_n)}f(v_n)\varphi$, then $\{{\tilde{w}}_n\}$ is bounded in $H_0^1(\mathrm{\Omega })$. Obviously, $〈J^{\prime }(v_n),{\tilde{w}}_n\varphi 〉\to 0$, i.e.

$$\begin{array}{r}-\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}[a{\int }_{\mathrm{\Omega }}\sqrt{1+2f^2(v_n)}f(v_n)\mathrm{\nabla }{v}_n\mathrm{\nabla }\varphi dx\\ +b\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)\left({\int }_{\mathrm{\Omega }}\frac{f(v_n)\mathrm{\nabla }{v}_n\mathrm{\nabla }\varphi }{\sqrt{1+2f^2(v_n)}}dx\right)]\\ =\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}\{a{\int }_{\mathrm{\Omega }}(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}){|\mathrm{\nabla }{v}_n|}^2\varphi dx+b\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)\\ \times \left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2\varphi }{1+2f^2(v_n)}dx\right)-{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)\varphi dx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)\varphi dx\}.\end{array}$$

(2.11)

On the other hand, by the Hölder inequality and (f₄) in Lemma 2.1, we have

$$\begin{array}{rcl}0& \le & \underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}|a{\int }_{\mathrm{\Omega }}\sqrt{1+2f^2(v_n)}f(v_n)\mathrm{\nabla }{v}_n\mathrm{\nabla }\varphi dx|\\ \le & C\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}|v_n\mathrm{\nabla }{v}_n\mathrm{\nabla }\varphi |dx\\ \le & C\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}\left[{\left({\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{v}_n|}^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_{\mathrm{\Omega }}{|v_n\mathrm{\nabla }\varphi |}^{2}dx\right)}^{\frac{1}{2}}\right]\\ \le & C\underset{ϵ\to 0}{lim}{\left({\int }_{B(x_j,2ϵ)}{|v|}^{2^{\ast }}dx\right)}^{\frac{1}{2^{\ast }}}=0.\end{array}$$

(2.12)

Similarly, we have

$$\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}\left[b\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)\left({\int }_{\mathrm{\Omega }}\frac{f(v_n)\mathrm{\nabla }{v}_n\mathrm{\nabla }\varphi }{\sqrt{1+2f^2(v_n)}}dx\right)\right]=0.$$

(2.13)

From the inequalities (2.11), (2.12), and (2.13), together with the following facts:

• (i)
  $$\begin{array}{rcl}\frac{1}{2}{|\mathrm{\nabla }{f}^2(v_n)|}^2\varphi & =& 2{|f(v_n)|}^2{|f^{\prime }(v_n)|}^2{|\mathrm{\nabla }{v}_n|}^2\varphi \\ =& \frac{2f^2(v_n)}{1+2f^2(v_n)}{|\mathrm{\nabla }{v}_n|}^2\varphi \\ \le & (1+\frac{2f^2(v_n)}{1+2f^2(v_n)}){|\mathrm{\nabla }{v}_n|}^2\varphi .\end{array}$$

• (ii) Similar to the proof of (2.12), it follows that

  $$\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)\varphi dx=0.$$

• (iii)
  $$\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{f}^2(v_n)|}^2\varphi dx={\mu }_j\text{and}\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)\varphi dx={\nu }_j.$$

We get

$$\begin{array}{rcl}0& =& \underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}\{a{\int }_{\mathrm{\Omega }}(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}){|\mathrm{\nabla }{v}_n|}^2\varphi dx+b\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)\\ \times \left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2\varphi }{1+2f^2(v_n)}dx\right)-{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)\varphi dx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)\varphi dx\}\\ \ge & \underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim}[\frac{a}{2}{\int }_{\mathrm{\Omega }}\varphi {|\mathrm{\nabla }{f}^2(v_n)|}^{2}dx-{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)\varphi dx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)\varphi dx]\\ =& \frac{a}{2}{\mu }_j-{\nu }_j.\end{array}$$

(2.14)

Combining this with Lemma 2.2, we obtain ${\nu }_j\ge 2^{-1}aS{\nu }_j^{\frac{2}{2^{\ast }}}$. This result implies that

$$(\mathrm{I}){\nu }_j=0\text{or}(\mathrm{II}){\nu }_j\ge {\left(2^{-1}aS\right)}^{\frac{N}{2}}.$$

If the second case ${\nu }_j\ge {(2^{-1}aS)}^{\frac{N}{2}}$ holds, for some $j\in I$, then by using the Hölder inequality, we have

$$\begin{array}{rcl}c& =& \underset{n\to \mathrm{\infty }}{lim}(J(v_n)-\frac{1}{4}〈J^{\prime }(v_n),v_n〉)\\ =& \underset{n\to \mathrm{\infty }}{lim}[\frac{a}{4}{\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx+(\frac{1}{4}-\frac{1}{2(2^{\ast })}){\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\\ -\lambda {\int }_{\mathrm{\Omega }}H(x,f(v_n))dx+\frac{\lambda }{4}{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)dx]\\ \ge & \underset{n\to \mathrm{\infty }}{lim}[\frac{1}{2N}{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx-\lambda {\int }_{\mathrm{\Omega }}H(x,f(v_n))dx+\frac{\lambda }{4}{\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)dx].\end{array}$$

By using inequality (2.4), we get

$$\begin{array}{rcl}c& =& \underset{n\to \mathrm{\infty }}{lim}(J(v_n)-\frac{1}{4}〈J^{\prime }(v_n),v_n〉)\\ \ge & \underset{n\to \mathrm{\infty }}{lim}[(\frac{1}{2N}-\lambda \epsilon ){\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx-\lambda c(\epsilon )|\mathrm{\Omega }|]\\ =& (\frac{1}{2N}-\lambda \epsilon )\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx-\lambda c(\epsilon )|\mathrm{\Omega }|.\end{array}$$

Since $0\le \varphi \le 1$, it follows that

$$c=\underset{n\to \mathrm{\infty }}{lim}(J(v_n)-\frac{1}{4}〈J^{\prime }(v_n),v_n〉)\ge (\frac{1}{2N}-\lambda \epsilon )\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)\varphi dx-\lambda c(\epsilon )|\mathrm{\Omega }|.$$

By using $f^{2(2^{\ast })}(v_n)\rightharpoonup f^{2(2^{\ast })}(v)$ in the measure sense and Lemma 2.2(i), we have

$$\begin{array}{rcl}c& =& \underset{n\to \mathrm{\infty }}{lim}(J(v_n)-\frac{1}{4}〈J^{\prime }(v_n),v_n〉)\\ \ge & (\frac{1}{2N}-\lambda \epsilon )\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v)\varphi dx+(\frac{1}{2N}-\lambda \epsilon )\sum _{j\in I}{\delta }_{x_j}(\varphi ){\nu }_j-\lambda c(\epsilon )|\mathrm{\Omega }|\\ \ge & (\frac{1}{2N}-\lambda \epsilon ){\nu }_j-\lambda c(\epsilon )|\mathrm{\Omega }|\\ \ge & \frac{1}{2N}{\left(2^{-1}aS\right)}^{\frac{N}{2}}-\lambda c\left(\frac{1}{4N\lambda }\right)|\mathrm{\Omega }|,\end{array}$$

where $\epsilon =1/4N\lambda$. This is impossible. Consequently, ${\nu }_j=0$ for all $j\in I$ and hence

$${\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\to {\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v)dx,\text{as }n\to +\mathrm{\infty }.$$

Thus, from the weak lower semicontinuity of the norm and $f\in C^{\mathrm{\infty }}$ we have

$$\begin{array}{rcl}o(1)\parallel v_n\parallel & =& 〈J^{\prime }(v_n),w_n〉=a{\int }_{\mathrm{\Omega }}(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}){|\mathrm{\nabla }{v}_n|}^{2}dx+b{\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)}^2\\ -\lambda {\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)dx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\\ =& a{\parallel v_n\parallel }^2+a{\int }_{\mathrm{\Omega }}\frac{2f^2(v_n)}{1+2f^2(v_n)}{|\mathrm{\nabla }{v}_n|}^{2}dx+b{\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }{v}_n|}^2}{1+2f^2(v_n)}dx\right)}^2\\ -\lambda {\int }_{\mathrm{\Omega }}h(x,f(v_n))f(v_n)dx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v_n)dx\\ \ge & a{\parallel v_n-v\parallel }^2+a{\parallel v\parallel }^2+a{\int }_{\mathrm{\Omega }}\frac{2f^2(v)}{1+2f^2(v)}{|\mathrm{\nabla }v|}^{2}dx+b{\left({\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }v|}^2}{1+2f^2(v)}dx\right)}^2\\ -\lambda {\int }_{\mathrm{\Omega }}h(x,f(v))f(v)dx-{\int }_{\mathrm{\Omega }}{f}^{2(2^{\ast })}(v)dx\\ =& a{\parallel v_n-v\parallel }^2+o(1)\parallel v\parallel ,\end{array}$$

since $J^{\prime }(v)=0$. Thus we prove that $\{v_n\}$ strongly converges to v in $H_0^1(\mathrm{\Omega })$. □

3 Existence of a sequence of arbitrarily small solutions

In this section, we prove the existence of infinitely many solutions of (1.1) which tend to zero. Let X be a Banach space and denote

$$\mathrm{\Sigma }:=\{A\subset X\setminus \{0\}:A\text{ is closed in }X\text{ and symmetric with respect to the origin}\}.$$

For $A\in \mathrm{\Sigma }$, we define genus $\gamma (A)$ as

$$\gamma (A):=inf\{m\in N:\mathrm{\exists }\phi \in C(A,R^m\setminus \{0\},-\phi (x)=\phi (-x))\}.$$

If there is no mapping φ as above for any $m\in N$, then $\gamma (A)=+\mathrm{\infty }$. Let ${\mathrm{\Sigma }}_k$ denote the family of closed symmetric subsets A of X such that $0\notin A$ and $\gamma (A)\ge k$. We list some properties of the genus (see [24], [35]).

Proposition 3.1

Let A and B be closed symmetric subsets of X which do not contain the origin. Then the following hold.

• (1) If there exists an odd continuous mapping from A to B, then $\gamma (A)\le \gamma (B)$.

• (2) If there is an odd homeomorphism from A to B, then $\gamma (A)=\gamma (B)$.

• (3) If $\gamma (B)<\mathrm{\infty }$, then $\gamma \overline{(A\setminus B)}\ge \gamma (A)-\gamma (B)$.

• (4) Then n-dimensional sphere $S^n$ has a genus of $n+1$ by the Borsuk-Ulam theorem.

• (5) If A is compact, then $\gamma (A)<+\mathrm{\infty }$ and there exists $\delta >0$ such that $U_{\delta }(A)\in \mathrm{\Sigma }$ and $\gamma (U_{\delta }(A))=\gamma (A)$, where $U_{\delta }(A)=\{x\in X:\parallel x-A\parallel \le \delta \}$.

The following version of the symmetric mountain-pass lemma is due to Kajikiya [24].

Lemma 3.1

Let E be an infinite-dimensional space and$J\in C^1(E,R)$and suppose the following conditions hold.

(C₁):$J(u)$is even, bounded from below, $J(0)=0$and$J(u)$satisfies the local Palais-Smale condition, i.e. for some$\overline{c}>0$, in the case when every sequence$\{u_k\}$in E satisfying${lim}_{k\to \mathrm{\infty }}J(u_k)=c<\overline{c}$and${lim}_{k\to \mathrm{\infty }}{\parallel J^{\prime }(u_k)\parallel }_{E^{\ast }}=0$has a convergent subsequence.

(C₂):For each$k\in N$, there exists an$A_k\in {\mathrm{\Sigma }}_k$such that${sup}_{u\in A_k}J(u)<0$.

Then either (R₁) or (R₂) below holds.

(R₁):There exists a sequence$\{u_k\}$such that$J^{\prime }(u_k)=0$, $J(u_k)<0$and$\{u_k\}$converges to zero.

(R₁):There exist two sequences$\{u_k\}$and$\{v_k\}$such that$J^{\prime }(u_k)=0$, $J(u_k)<0$, $u_k\ne 0$, ${lim}_{k\to \mathrm{\infty }}{u}_k=0$, $J^{\prime }(v_k)=0$, $J(v_k)<0$, ${lim}_{k\to \mathrm{\infty }}{v}_k=0$, and$\{v_k\}$converges to a non-zero limit.

Remark 3.1

From Lemma 3.1 we have a sequence $\{u_k\}$ of critical points such that $J(u_k)\le 0$, $u_k\ne 0$ and ${lim}_{k\to \mathrm{\infty }}{u}_k=0$.

In order to get infinitely many solutions we need some lemmas. Let $\epsilon =\frac{1}{2(2^{\ast })\lambda }$, from (2.4) we have

$$\begin{array}{rcl}J(v)& :=& \frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx+\frac{b}{4}{\left({\int }_{\mathrm{\Omega }}{|f^{\prime }(v)|}^2{|\mathrm{\nabla }v|}^{2}dx\right)}^2-\frac{1}{2(2^{\ast })}{\int }_{\mathrm{\Omega }}{|f(v)|}^{2(2^{\ast })}dx\\ -\lambda {\int }_{\mathrm{\Omega }}H(x,f(v))dx\\ \ge & \frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx-(\frac{1}{2(2^{\ast })}+\epsilon \lambda ){\int }_{\mathrm{\Omega }}{|f(v)|}^{2(2^{\ast })}dx-\lambda b(\epsilon )|\mathrm{\Omega }|\\ =& \frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx-\frac{1}{2^{\ast }}{\int }_{\mathrm{\Omega }}{|f(v)|}^{2(2^{\ast })}dx-\lambda b\left(\frac{1}{2(2^{\ast })\lambda }\right)|\mathrm{\Omega }|\\ \ge & L_1{\parallel v\parallel }^2-L_2{\parallel v\parallel }^{2^{\ast }}-L_3\lambda ,\end{array}$$

where $L_1$, $L_2$, $L_3$ are some positive constants.

Let $Q(t)=L_1{t}^2-L_2{t}^{2^{\ast }}-L_3\lambda$. Then

$$J(v)\ge Q(\parallel v\parallel ).$$

Furthermore, there exists ${\lambda }_{\ast }:=\frac{2L_1}{N{L}_3}{(\frac{2L_1}{2^{\ast }{L}_2})}^{(N-2)/2}$ such that for $\lambda \in (0,{\lambda }_{\ast })$, $Q(t)$ attains its positive maximum, that is, there exists

$$R_1={\left(\frac{2L_1}{2^{\ast }{L}_2}\right)}^{(N-2)/4}$$

such that

$$e_1=Q(R_1)=\underset{t\ge 0}{max}Q(t)>0.$$

Therefore, for $e_0\in (0,e_1)$, we may find $R_0<R_1$ such that $Q(R_0)=e_0$. Now we define

$$\chi (t)=\{\begin{array}{ll}1,& 0\le t\le R_0,\\ \frac{L_1{t}^2-\lambda L_3-e_1}{L_2{t}^{2^{\ast }}},& t\ge R_1,\\ C^{\mathrm{\infty }},\chi (t)\in [0,1],& R_0\le t\le R_1.\end{array}$$

Then it is easy to see $\chi (t)\in [0,1]$ and $\chi (t)$ is $C^{\mathrm{\infty }}$. Let $\phi (v)=\chi (\parallel v\parallel )$ and consider the perturbation of $J(v)$:

$$\begin{array}{rcl}G(v)& :=& \frac{a}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx+\frac{b}{4}{\left({\int }_{\mathrm{\Omega }}{|f^{\prime }(v)|}^2{|\mathrm{\nabla }v|}^{2}dx\right)}^2-\frac{1}{2(2^{\ast })}\phi (v){\int }_{\mathrm{\Omega }}{|f(v)|}^{2(2^{\ast })}dx\\ -\lambda \phi (v){\int }_{\mathrm{\Omega }}H(x,f(v))dx.\end{array}$$

(3.1)

Then

$$G(v)\ge L_1{\parallel v\parallel }^2-L_2\phi (v){\parallel v\parallel }^{2^{\ast }}-L_3\lambda =\overline{Q}(\parallel v\parallel ),$$

where $\overline{Q}(t)=L_1{t}^2-L_2\chi (t)t^{2^{\ast }}-L_3\lambda$ and

$$\overline{Q}(t)=\{\begin{array}{ll}Q(t),& 0\le t\le R_0,\\ e_1,& t\ge R_1.\end{array}$$

From the above arguments, we have the following.

Lemma 3.2

Let$G(v)$is defined as in (3.1). Then

• (i) $G\in C^1(H_0^1(\mathrm{\Omega }),R)$ and G is even and bounded from below.

• (ii) If $G(v)<e_0$, then $\overline{Q}(\parallel v\parallel )<e_0$, consequently, $\parallel v\parallel <R_0$ and $I(v)=G(v)$.

• (iii) There exists ${\lambda }^{\ast }$ such that, for $\lambda \in (0,{\lambda }^{\ast })$, G satisfies a local (PS) _c condition for

  $$c<e_0\in (0,min\{e_1,\frac{1}{2N}{\left(2^{-1}aS\right)}^{\frac{N}{2}}-\lambda c\left(\frac{1}{4N\lambda }\right)|\mathrm{\Omega }|\}).$$

Lemma 3.3

Assume that (H₃) of Theorem  1.1holds. Then for any$k\in N$, there exists$\delta =\delta (k)>0$such that$\gamma (\{v\in H_0^1(\mathrm{\Omega }):G(v)\le -\delta (k)\}\setminus \{0\})\ge k$.

Proof

Firstly, by (H₃) of Theorem 1.1, for any fixed $v\in D_0^{1,2}(\mathrm{\Omega })$, $v\ne 0$, we have

$$H(x,\rho v)\ge M(\rho ){(\rho v)}^2\text{with }M(\rho )\to \mathrm{\infty }\text{ as }\rho \to 0.$$

Secondly, given any $k\in N$, let $E_k$ be a k-dimensional subspace of $H_0^1(\mathrm{\Omega })$. Then there exists a constant ${\sigma }_k$ such that

$$\parallel v\parallel \le {\sigma }_k{|v|}_2,\mathrm{\forall }v\in E_k.$$

Therefore for any $v\in E_k$ with $\parallel v\parallel =1$ and ρ small enough, by (f₁) in Lemma 2.1 we have

$$\begin{array}{rcl}G(\rho v)& =& \frac{a{\rho }^2}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx+\frac{b{\rho }^4}{4}{\left({\int }_{\mathrm{\Omega }}{|f^{\prime }(\rho v)|}^2{|\mathrm{\nabla }v|}^{2}dx\right)}^2\\ -\frac{1}{2(2^{\ast })}\phi (\rho v){\int }_{\mathrm{\Omega }}{|f(\rho v)|}^{2(2^{\ast })}dx-\lambda \phi (\rho v){\int }_{\mathrm{\Omega }}H(x,f(\rho v))dx\\ \le & \frac{a{\rho }^2}{2}+\frac{b{\rho }^4}{4}-\frac{\lambda M(\rho )\phi (\rho v)}{{\sigma }_k^2}{\rho }^2\\ \le & (\frac{a}{2}+\frac{b{\rho }^2}{4}-\frac{\lambda M(\rho )\phi (\rho v)}{{\sigma }_k^p}){\rho }^2\\ =& -\delta (k)<0,\end{array}$$

since ${lim}_{|\rho |\to 0}M(\rho )=+\mathrm{\infty }$. That is,

$$\{v\in E_k:\parallel v\parallel =\rho \}\subset \{v\in H_0^1(\mathrm{\Omega }):G(v)\le -\delta (k)\}\setminus \{0\}.$$

This completes the proof. □

Now we give the proof of Theorem 1.1.

Proof of Theorem 2.1

Recall that

$${\mathrm{\Sigma }}_k=\{A\in H_0^1(\mathrm{\Omega })\setminus \{0\}:A\text{ is closed and }A=-A,\gamma (A)\ge k\}$$

and define

$$c_k=\underset{A\in {\mathrm{\Sigma }}_k}{inf}\underset{u\in A}{sup}G(v).$$

By Lemmas 3.2(i) and Lemmas 3.3, we know that $-\mathrm{\infty }<c_k<0$. Therefore, assumptions (C₁) and (C₂) of Lemma 3.1 are satisfied. This means that G has a sequence of solutions $\{v_n\}$ converging to zero. Hence, Theorem 2.1 follows by Lemma 3.2(ii). □

Proof of Theorem 1.1

This follows from Theorem 2.1, since $u_m=f(v_m)\ne u_n=f(v_n)$ if $v_m\ne v_n$ and $f\in C^{\mathrm{\infty }}$. □