Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in $\mathbb{R}^{N}$

TL;DR

This paper establishes the existence and multiplicity of solutions for fourth-order Kirchhoff type elliptic equations with concave-convex nonlinearities in , using variational methods and covering cases with  and small positive .

Contribution

It extends known results by proving solution existence and multiplicity for a broad class of Kirchhoff equations with concave-convex nonlinearities, including the case .

Findings

Existence of at least one solution for .

At least two solutions for small positive .

Infinitely many solutions if the nonlinearity is odd in u.

Abstract

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} \Delta^{2}u-M(\|\nabla u\|_{2}^{2})\Delta u+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N}, \end{eqnarray*} where $M(t):\mathbb{R}\rightarrow\mathbb{R}$ is the Kirchhoff function, $f(x,u)=\lambda k(x,u)+ h(x,u)$, $\lambda\geq0$, $k(x,u)$ is of sublinear growth and $h(x,u)$ satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for $\lambda=0$. For $\lambda>0$ small enough, we obtain at least two nontrivial solutions. Furthermore, if $f(x,u)$ is odd in $u$, we show that above equations possess infinitely many solutions for all $\lambda\geq0$. Our theorems generalize some known results in the literatures even for $\lambda=0$ and our proof is based on the variational methods.

Equations

$${\Delta }^{2}u-M(\Vert \nabla u{\Vert }_2^2)\Delta u+V(x)u=f(x,u),x\in {\mathbb{R}}^N,$$

$$\displaystyle\left\{\begin{array}[]{ll}-\left(a+b\displaystyle\int_{\Omega}|\nabla u|^{2}dx\right)\Delta u=f(x,u)&\mbox{in $\Omega$},\\ u(x)=0,&\mbox{on $\partial\Omega$},\end{array}\right.$$

$$u_{tt}-\left(a+b{\int }_{\Omega }|\nabla u{|}^{2}dx\right)\Delta u=g(x,u).$$

$$\rho \frac{{\partial }^{2}u}{{\partial }^{2}t}-\left(\frac{P_0}{h}+\frac{E}{2L}{\int }_0^L|\frac{\partial u}{\partial x}{|}^{2}dx\right)\frac{{\partial }^{2}u}{{\partial }^{2}x}=g(x,u)$$

$$f(x,u)=\varpi s_1(x)|u{|}^{{\tau }_1-2}u+s_2(x)|u{|}^{{\tau }_2-2}u.$$

$$u_{tt}+{\Delta }^{2}u-\left(a+b{\int }_{\Omega }|\nabla u{|}^{2}dx\right)\Delta u=f(x,u).$$

$$\nu \tilde{F}(x,t)\ge \tilde{F}(x,st)$$

$$f(x,t)t\ge \mu F(x,t),\text{ for a.e. }x\in {\mathbb{R}}^N\text{ and }\forall |t|\ge l_1.$$

$$s_0|t{|}^{2\iota }\tilde{F}(x,t)\ge |F(x,t){|}^{\iota }\text{ for a.e. }x\in {\mathbb{R}}^N\text{ and }\forall |t|\ge l_2.$$

$$|w(x,t)|\le C(1+|t{|}^{p-1})\text{for some }4<p<2^{\ast }=\{\begin{array}{ll}\frac{2N}{N-4},& 8>N>4\\ +\infty ,& 1<N\le 4,\end{array}$$

$$\frac{1}{4}w(x,t)t-W(x,t)\ge -\frac{d}{4}|t{|}^2,\text{ for a.e. }x\in {\mathbb{R}}^N\text{ and }\forall |t|\ge L.$$

$$\underset{|y|\to \infty }{\lim }meas\left\{x\in {\mathbb{R}}^N:|x-y|\le \bar{r},V(x)\le \mathcal{M}\right\}=0,\forall \mathcal{M}>0.$$

$${\sigma }_2(t^2+t)\ge \hat{M}(t)\ge \frac{1}{2}M(t)t+{\sigma }_{1}t\text{mbox}foranyt\ge 0,$$

$$\displaystyle\widehat{M}(t)=\frac{at+bt^{2}}{1+\ln(t^{2}+1)}\ \ \mbox{with $a,b>0$}.$$

$$f(x,t)=\lambda k(x,t)+h(x,t).$$

$$|k(x,t)|\le b_1(x)|t{|}^{r_1-1}+b_2(x)|t{|}^{r_2-1},$$

$$\tilde{H}(x,t)=\frac{1}{4}h(x,t)t-H(x,t)\ge -d_1{t}^2\text{mbox}forall\text{ a.e. }x\in {\mathbb{R}}^N\text{mbox}and|t|\ge {\rho }_{\infty };$$

$$|h(x,t)|\le d_2(|t|+|t{|}^{\zeta -1})\text{mbox}forall\text{ a.e. }x\in {\mathbb{R}}^N\text{mbox}andt\in \mathbb{R}.$$

$$|I(x_j)|<\infty \text{mbox}and{I}^{\prime }{|}_{E_j}(x_j)\to 0,$$

$$I(u)\le 0\text{mbox}on{\tilde{X}}_m\setminus B_{r_m},m=1,2,\cdots .$$

$$H^2({\mathbb{R}}^N)=\left\{u\in L^2({\mathbb{R}}^N):\nabla u\in L^2({\mathbb{R}}^N),\Delta u\in L^2({\mathbb{R}}^N)\right\}$$

$$\Vert u{\Vert }_{H^2}^2={\int }_{{\mathbb{R}}^N}(|\Delta u{|}^2+|\nabla u{|}^2+u^2)dx$$

$$E=\left\{u\in H^2({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^N}(|\Delta u{|}^2+|\nabla u{|}^2+V(x)u^2)dx<+\infty \right\}$$

$$⟨u,v{⟩}_E={\int }_{{\mathbb{R}}^N}(\Delta u\cdot \Delta v+\nabla u\cdot \nabla v+V(x)uv)dx$$

$$\Vert u{\Vert }_s\le C_s\Vert u{\Vert }_E\text{mbox}forallu\in E.$$

$$I(u)$$

$$⟨I^{\prime }(u),v⟩$$

$$|K(t,x)|\le \frac{1}{r_1}{b}_1(x)|t{|}^{r_1}+\frac{1}{r_2}{b}_2(x)|t{|}^{r_2}$$

$$|H(x,t)|\le \epsilon |t{|}^2+D_{\epsilon }|t{|}^{\zeta },\forall (x,t)\in {\mathbb{R}}^N\times \mathbb{R}.$$

Full text

**Solutions for fourth-order

Kirchhoff type elliptic equations involving concave-convex nonlinearities in $\mathbb{R}^{N}$** ††thanks: This work is supported by NSF of China (No.11801472) and NSF of China (No.11701463) and China Scholarship Council (No.201708515186) and Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No.18TD0013), Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No.2017CXTD02).

Dong-Lun Wua,b111Corresponding author., Fengying Lic

aCollege of Science, Southwest Petroleum University,

Chengdu, Sichuan 610500, P.R. China

bInstitute of Nonlinear Dynamics, Southwest Petroleum University,

Chengdu, Sichuan 610500, P.R. China

cThe School of Economic and Mathematics, Southwestern University of Finance and Economics,

Chengdu 611130, P. R. China

11footnotetext: E-mail: wudl2008@163.com

Abstract In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations

[TABLE]

where $M(t):\mathbb{R}\rightarrow\mathbb{R}$ is the Kirchhoff function, $f(x,u)=\lambda k(x,u)+h(x,u)$, $\lambda\geq 0$, $k(x,u)$ is of sublinear growth and $h(x,u)$ satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for $\lambda=0$. For $\lambda>0$ small enough, we obtain at least two nontrivial solutions. Furthermore, if $f(x,u)$ is odd in $u$, we show that above equations possess infinitely many solutions for all $\lambda\geq 0$. Our theorems generalize some known results in the literatures even for $\lambda=0$ and our proof is based on the variational methods.

Keywords Fourth-order Kirchhoff type elliptic equations; Concave-convex nonlinearities; General growth conditions; Variational methods.

1 Introduction

In this paper, we study the existence of multiple solutions for the following fourth-order Kirchhoff type elliptic equations

[TABLE]

where $M\in C([0,+\infty),\mathbb{R})$ is a Kirchhoff-type function, potential $V(x)\in C(\mathbb{R},\mathbb{R}^{+})$ and nonlinearities $f(x,u)\in C(\mathbb{R}^{N}\times\mathbb{R},\mathbb{R})$, $1<N<8$. The Kirchhoff type problems on a bounded domain is introduced as

[TABLE]

which is related to the stationary analogue of the Kirchhoff equation

[TABLE]

Equation (5) was proposed by Kirchhoff in 1883 as a generalization of following d’Alembert’s wave equation

[TABLE]

for free vibrations of elastic strings. It well known that, as a useful model, the Kirchhoff equation has many applications in mechanical and biological problems. After the work of Lions [12], the existence and multiplicity of solutions for Kirchhoff equations have been studied by many mathematicians via the variational methods. In recent years, some authors considered the Kirchhoff equations or the $p$-Kirchhoff equations with concave-convex nonlinearities. We remained the readers with references [3, 5, 6, 7, 10, 13, 14, 17, 24, 27]. However, the nonlinearities were required to satisfy some specific form in these papers, such as

[TABLE]

In the present paper, we study the concave-convex nonlinearities with abstract forms.

Subsequently, we recall some known results about the fourth-order Kirchhoff type elliptic equations which has been studied by many mathematicians [1, 8, 9, 11, 15, 16, 20, 25, 26]. In 2012, Wang and An [19] considered problem (1) in a bounded domain with potential $V(x)\equiv 0$ and $M(t)\equiv const.$ when $t$ large enough. Under some 2-superlinear conditions, the authors obtain a nonnegative solution by using the Mountain Pass Theorem. This problem is related to the stationary analog of the evolution equation of Kirchhoff type

[TABLE]

Actually, if $M(t)=a+bt$, we can obtain solutions for problem (1) when the growth of the nonlinearities is required to be 4-superlinear which has been shown by some previous works. Whether there are solutions for problem (1) with 2-superlinear nonlinearities when $M(t)=a+bt$ is still open. In this paper, we only consider the 4-superlinear case. In order to study different 4-superlinear nonlinearities, some different kinds of growth conditions were introduced. In [16], Song and Chen showed problem (1) possesses infinitely many solutions under the following monotonous condition.

$(MC)$ There exists $\nu\geq 1$ such that

[TABLE]

for all $(x,t)\in\mathbb{R}^{N}\times\mathbb{R}$ and $s\in[0,1]$, where $\widetilde{F}(x,t)=\frac{1}{4}f(x,t)t-F(x,t)$ and $F(x,t)=\int_{0}^{t}f(x,s)ds$.

By replacing $(MC)$ with the following local $(AR)$-type condition, Song and Chen [16] also obtained infinitely many solutions for problem (1).

$(AR)$ There exist $l_{1}>0$ and $\mu>4$ such that

[TABLE]

In 2015, Xu and Chen [26] considered problem (1) in $\mathbb{R}^{3}$ and obtained infinitely many solutions under the following superlinear condition which is weaker than $(AR)$.

$(WAR)$ There exist constants $s_{0}$, $l_{2}>0$ and $\iota>\frac{3}{2}$ such that

[TABLE]

Obviously, we can obtain the following condition with $(MC)$, $(AR)$ and $(WAR)$ respectively.

$(FSL)$ $\widetilde{F}(x,t)\geq 0$ for a.e. $x\in\mathbb{R}^{N}$ and $|t|$ large enough.

In a recent paper, Ding and Li [8] considered a class of nonhomogenous fourth-order Kirchhoff equations with $(FSL)$. They obtained the following theorem.

Theorem 1.1**.**

(See [8]) Assume that $f(x,t)=w(x,t)+g(x)$, where $g\in L^{2}(\mathbb{R}^{N})$, $g\not\equiv 0$ and the following conditions hold.

• $(V^{\prime})$

$V\in C(\mathbb{R}^{N},\mathbb{R})$* satisfies $\displaystyle\inf_{x\in\mathbb{R}^{N}}V(x)\geq V_{0}>0$ and for each $M>0,$ meas*$\{x\in\mathbb{R}^{N}:V(x)\leq M\}<+\infty$, where $V_{0}$ is a constant and meas denotes the Lebesgue measure in $\mathbb{R}^{N}$.

• $(w_{1})$

$w\in C(\mathbb{R}^{N}\times\mathbb{R},\mathbb{R})$* and*

[TABLE]

where $C$ is a positive constant.

• $(w_{2})$

$w(x,t)=o(|t|)$* as $|t|\to 0$ uniformly in $x\in\mathbb{R}^{N}$.*

• $(w_{3})$

$\frac{W(x,t)}{t^{4}}\to+\infty$* as $|t|\to+\infty$ uniformly in $x\in\mathbb{R}^{N}$, where $W(x,t)=\int_{0}^{t}w(x,s)ds$..*

• $(w_{4})$

There exist $L>0$ and $d\in\left[0,\frac{V_{0}}{2}\right]$ such that

[TABLE]

Then, there exists a constant $g_{0}>0$ such that the problem (1) has at least two different solutions whenever $\|g\|_{L^{2}}<g_{0}$ and $M(t)=a+bt$ with $a,b>0$, one is negative energy solution, and the other is positive energy solution.

Remark 1**.**

The condition $(w_{4})$ is required to hold for any $t\geq L$ in [8]. However, the authors used this condition for any $|t|\geq L$ implicitly.

In order to use the variational methods to obtain the results, it is not enough to show the geometric structure of the corresponding functional. We also need to guarantee the convergence of the asymptotic critical sequence which can be obtained by the compactness of the embedding. However, since the domain is unboundedness, there is no natural compact embedding to use. To overcome this difficulty, periodic, coercive and radial symmetric conditions are put forward. Condition $(V^{\prime})$ is a classical coercive condition on $V(x)$ to make sure the embedding is compact. It has been shown by Bartsch and Wang in [2] that the following coercive condition is weaker than $(V^{\prime})$.

$(V)$ $V\in C(\mathbb{R}^{N},\mathbb{R})$, $\inf_{x\in\mathbb{R}^{N}}V(x)>0$. There exists $\bar{r}>0$ such that

[TABLE]

There are still some other ways to get the compactness back(see[28, 21, 23, 22]). In this paper, we consider the coercive case and use condition $(V)$ to obtain the compactness of the embedding.

Before we state our results, we introduce some conditions on $M$. In problem (1), the Kirchhoff function $M(t)$ is a abstract function, which has been rarely considered when the problem lies in $\mathbb{R}^{N}$. Through out this paper, we assume that $M(t)$ satisfies the following conditions.

$(M_{1})$ $M\in C([0,+\infty),\mathbb{R})$ and there exists $m_{0}>0$ such that $M(t)\geq m_{0}$ for all $t\in[0,+\infty)$.

$(M_{2})$ There exist positive constants $\sigma_{1}$ and $\sigma_{2}$ such that

[TABLE]

where $\widehat{M}(t)=\int_{0}^{t}M(s)ds$.

Remark 2**.**

It is easy to check that the original Kirchhoff function $M(t)=a+bt$ for any $a,b>0$ satisfies $(M_{1})$-$(M_{2})$. There are still some other functions admitting our conditions, such as

[TABLE]

Subsequently, in order to study the concave-convex nonlinearities, we consider

[TABLE]

Letting $K(x,t)=\int^{t}_{0}k(x,v)dv$ and $H(x,t)=\int^{t}_{0}h(x,v)dv$, we state our main theorems.

Theorem 1.2**.**

Suppose that (6), $(V)$, $(M_{1})$, $(M_{2})$ and the following conditions hold

• $(f_{1})$

there exit $\bar{x}\in\mathbb{R}^{N}$, $r_{0}\in(1,2)$ and $b_{0}>0$ such that $K(\bar{x},t)\geq b_{0}|t|^{r_{0}}$ for all $t\in\mathbb{R}$;

• $(f_{2})$

for any $(x,t)\in\mathbb{R}^{N}\times\mathbb{R}$, there exist $r_{1}$, $r_{2}\in(1,2)$ such that

[TABLE]

where $b_{i}(x)\in L^{\beta_{i}}(\mathbb{R}^{N},\mathbb{R}^{+})$ and $\beta_{i}\in\left(\frac{2^{*}}{2^{*}-r_{i}},\frac{2}{2-r_{i}}\right]$ for $i=1,2$;

• $(f_{3})$

$h(x,t)=o(|t|)$* as $t\rightarrow 0$ uniformly in $x\in\mathbb{R}^{N}$;*

• $(f_{4})$

$H(x,t)/t^{4}\rightarrow+\infty$* as $|t|\rightarrow\infty$ uniformly in $x\in\mathbb{R}^{N}$;*

• $(f_{5})$

there exist positive constants $d_{1}$ and $\rho_{\infty}$ such that

[TABLE]

• $(f_{6})$

there exist $d_{2}>0$ and $4<\zeta<2^{*}$ such that

[TABLE]

Then problem (1) possesses at least one solution for $\lambda=0$ and there exists $\lambda_{1}>0$ such that for any $\lambda\in(0,\lambda_{1})$, problem (1) possesses at least two solutions.

Theorem 1.3**.**

Suppose that (6), $(V)$, $(M_{1})$, $(M_{2})$, $(f_{1})$-$(f_{6})$ and

$(f_{7})$* $K(x,-t)=K(x,t)$ and $H(x,-t)=H(x,t)$ for all $(x,t)\in\mathbb{R}^{N}\times\mathbb{R}$.*

Then for any $\lambda\geq 0$, problem (1) possesses infinitely many solutions.

Remark 3**.**

Obviously, condition $(f_{5})$ is weaker than $(w_{4})$. We can also see that $(f_{5})$ is weaker than $(MC)$, $(AR)$ and $(WAR)$. Hence Theorem 1.2 and 1.3 generalize Theorems 1.1, 1.2, 1.3 in [16] and Theorems 1.1, 1.2 in [26].

Remark 4**.**

In Theorems 1.2 and 1.3, the sign of $H(x,t)$ is indefinite. Although we have $(f_{4})$, $H(x,t)$ can also be negative around origin with respect to $t$.

Remark 5**.**

Although there were some papers concerning on the fourth-order Kirchhoff type elliptic equations with concave-convex nonlinearities on bounded domain, to the best of the knowledge of the authors, this is the first work on fourth-order Kirchhoff type elliptic equations with concave-convex growth on unbounded domain.

In this paper, we will use the variational method to prove our theorems. First, we introduce the definition of the $(PS)^{*}$ condition.

Definition 1**.**

Let $E$ be a Hilbert space. A functional $I\in C^{1}(E,R)$ is said to satisfy the $(PS)^{*}$ condition with respect to $E_{j}$, $j=1,2,\cdots$, if any sequence $x_{j}\in E_{j}$ satisfying

[TABLE]

imply a convergent subsequence, where $E_{j}$ is a sequence of linear subspace of $E$ with finite dimensional.

The following critical point theorem is needed to obtain the multiplicity of solutions.

Lemma 1.1**.**

(Chang[4]) Suppose that $E$ is a Hilbert space, $I\in C^{1}(E,\mathbb{R})$ is even with $I(0)=0$, and that

• $(C_{1})$

there exist $\varrho$, $\alpha>0$ and a finite dimensional linear subspace $X$ such that $I|_{X^{\bot}\bigcap\partial B_{\varrho}}\geq\alpha$, where $B_{\varrho}=\{u\in E:\|u\|_{E}\leq\varrho\}$;

• $(C_{2})$

there is a sequence of linear subspaces $\tilde{X}_{m}$, $dim\tilde{X}_{m}=m$, and there exists $r_{m}>0$ such that

[TABLE]

If, further, $I$ satisfies the $(PS)^{*}$ condition with respect to $\{\tilde{X}_{m}|m=1,2,\cdots\}$, then $I$ possesses infinitely many distinct critical points corresponding to positive critical values.

2 Preliminaries

In this paper, we let

[TABLE]

with the norm

[TABLE]

Set

[TABLE]

with the inner product

[TABLE]

and the norm $\|u\|_{E}=\langle u,u\rangle^{1/2}$. Obviously, It is well known that under hypothesis $(V)$, the embedding $E\hookrightarrow L^{s}(\mathbb{R}^{N})$ is continuous for $s\in[2,2^{*}]$ and compact for $s\in[2,2^{*})$. Then, for any $s\in[2,2^{*}]$, there exists $C_{s}>0$ such that

[TABLE]

It is known that the weak solutions for problem (1) are the critical points of the following functional

[TABLE]

Similar to the proof of Proposition 2.2 in [22], under $(f_{2})$ and $(f_{3})$, we see that $I\in C^{1}(E,\mathbb{R})$ and for each $u$, $v\in E$,

[TABLE]

3 Proof of Theorem 1.2

Lemma 3.1**.**

Assume (6), $(V)$, $(M_{1})$, $(M_{2})$, $(f_{2})$, $(f_{3})$ and $(f_{6})$ hold, then there exists $\lambda_{1}>0$ such that for all $\lambda\in[0,\lambda_{1})$, there exist $\varrho$, $\alpha>0$ such that $I|_{\partial B_{\varrho}}\geq\alpha$ , where $B_{\varrho}=\{u\in E:\|u\|_{E}\leq\varrho\}$.

Proof. By $(f_{2})$, we obtain that

[TABLE]

for all $(x,t)\in\mathbb{R}^{N}\times\mathbb{R}$. It follows from $(f_{3})$ and $(f_{6})$, for any $\varepsilon>0$, there exists $D_{\varepsilon}>0$ such that

[TABLE]

By (7), $(M_{1})$, $(M_{2})$, (9) and (10), we have

[TABLE]

where $M_{1}=\max\left\{\frac{1}{r_{1}}C_{r_{1}\beta_{1}^{*}}^{r_{1}}\|b_{1}\|_{\beta_{1}},\frac{1}{r_{2}}C_{r_{2}\beta^{*}}^{r_{2}}\|b_{2}\|_{\beta}\right\}$. If we choose $\varepsilon$ small enough, it is easy to see that there exist positive constants $\lambda_{1}$, $\varrho$ and $\alpha$ such that $I|_{\partial B_{\varrho}}\geq\alpha$ for all $\lambda\in[0,\lambda_{1})$. We finish the proof of this lemma.

Lemma 3.2**.**

Suppose (6), $(V)$, $(M_{2})$, $(f_{2})$ and $(f_{4})$ hold, then there exists $\tilde{e}\in E$ such that $\|\tilde{e}\|>\varrho$ and $I(\tilde{e})\leq 0$, where $\varrho$ is defined in Lemma 3.1.

Proof. Choose $e\in C^{\infty}_{0}(\Upsilon_{1}(0),\mathbb{R})$ such that $\|e\|_{E}=1$, where $\Upsilon_{r}(x_{0})=\{x\in\mathbb{R}^{N}:|x-x_{0}|\leq r\}$. We can see that there exist $L_{0}>0$ and $\Sigma\subset\Upsilon_{1}(0)$ such that $|e|\geq L_{0}$ for all $x\in\Sigma$ with $meas(\Sigma)>0$. By $(f_{4})$, for any $A>0$ there exists $Q>0$ such that

[TABLE]

for all $|t|\geq Q$ and $x\in\mathbb{R}^{N}$, which implies that

[TABLE]

for all $\xi\geq Q/L_{0}$. By (7), $(M_{2})$, (9) and (11), for any $\xi>0$ large enough, we have

[TABLE]

By the arbitrariness of $A$, there exists $\xi_{0}>0$ such that $I(\xi_{0}e)<0$ and $\|\xi_{0}e\|>\varrho$. Let $\tilde{e}=\xi_{0}e$, we can see $I(\tilde{e})<0$, which proves this lemma.

Lemma 3.3**.**

Suppose (6), $(V)$, $(M_{1})$, $(M_{2})$ and $(f_{2})$-$(f_{6})$ hold, then $I$ satisfies the $(PS)$ condition.

Proof. Let $\{u_{n}\}\subset E$ be a sequence such that $\{I(u_{n})\}$ is bounded and $I^{\prime}(u_{n})\rightarrow 0$ as $n\rightarrow\infty$. Then there exists a constant $M_{2}>0$ such that

[TABLE]

Next, we show that $\{u_{n}\}$ is bounded in $E$. Arguing in an indirect way, we assume that $\|u_{n}\|_{E}\rightarrow+\infty$ as $n\rightarrow\infty$. Set $z_{n}=\frac{u_{n}}{\|u_{n}\|_{E}}$, then $\|z_{n}\|_{E}=1$, which implies that there exists a subsequence of $\{z_{n}\}$, still denoted by $\{z_{n}\}$, such that $z_{n}\rightharpoonup z_{0}$ in $E$ and $z_{n}\rightarrow z_{0}$ a.e. in $\mathbb{R}^{N}$ as $n\rightarrow\infty$. From $(f_{3})$, we can deduce that $\widetilde{H}(x,t)=o(t^{2})$ as $|t|\rightarrow 0$ uniformly in $x$, then there exists $\rho_{0}\in(0,\rho_{\infty})$ such that

[TABLE]

for all $|t|\leq\rho_{0}$ and $x\in\mathbb{R}^{N}$ . If $z_{0}\equiv 0$, we can deduce

[TABLE]

which is a contradiction. Then we have $z_{0}\not\equiv 0$. Let $\Omega=\{x\in\mathbb{R}^{N}|\ |z_{0}(x)|>0\}$. Then we can see that $meas(\Omega)>0$. Since $\|u_{n}\|_{E}\rightarrow+\infty$ as $n\rightarrow\infty$ and $|u_{n}|=|z_{n}|\cdot\|u_{n}\|_{E}$, then we have $|u_{n}|\rightarrow+\infty$ as $n\rightarrow\infty$ for a.e. $x\in\Omega$. It follows from $(f_{3})$ and $(f_{4})$ that there exists $M_{3}>0$ such that

[TABLE]

for all $(x,t)\in\mathbb{R}\times\mathbb{R}^{N}$. Hence, by (7), we obtain

[TABLE]

which implies

[TABLE]

Moreover, we deduce from $(f_{4})$ and Fatou’s Lemma that

[TABLE]

It follows from $(M_{2})$ that

[TABLE]

which implies that

[TABLE]

which is a contradiction. Hence $\{u_{n}\}$ is bounded in $E$. Then there exists a subsequence, still denoted by $\{u_{n}\}$, such that $u_{n}\rightharpoonup u$ in $E$. Therefore

[TABLE]

By (7) and $(f_{2})$, we have

[TABLE]

For $i=1,2$, set

[TABLE]

It is easy to see that $\left(\frac{2^{*}}{2^{*}-r_{i}},\frac{2}{2-r_{i}}\right]=S_{i,1}\bigcup S_{i,2}$ and $S_{i,1}\bigcap S_{i,2}\neq\emptyset$. Moreover, let

[TABLE]

By an easy computation, we deduce that there exists $\eta_{i}\in[2,2^{*})$ such that $\frac{1}{\beta_{i}}+\frac{r_{i}-1}{\xi_{i}}+\frac{1}{\eta_{i}}=1$. It follows from (7) and $(f_{2})$ that

[TABLE]

Therefore, $\int_{\mathbb{R}^{N}}(f(x,u_{n})-f(x,u))(u_{n}-u)dx\rightarrow 0$ as $n\rightarrow\infty$. It follows from (8) that

[TABLE]

Define a linear functional $\mathcal{B}:E\rightarrow\mathbb{R}$ as

[TABLE]

It can be deduced that $\mathcal{B}$ is continuous on $E$. Since $u_{n}\rightharpoonup u$ in $E$, we obtain that

[TABLE]

Hence, by the boundedness of $\{u_{n}\}$ and the continuousness of $M(t)$, we have

[TABLE]

which implies that $\|u_{n}-u\|_{E}\rightarrow 0$ as $n\rightarrow\infty$. Hence $I$ satisfies the $(PS)$ condition.

From Lemmas 3.1-3.3 and the Mountain Pass Theorem, for any $\lambda\in[0,\lambda_{1})$, we can obtain a critical point $u^{*}$ of $I$ satisfying $I(u^{*})\geq\alpha$ and $I^{\prime}(u^{*})=0$. The following lemma tell us that there exists another nontrivial critical point of $I$ corresponding to negative critical value.

Lemma 3.4**.**

Suppose that (6), $(V)$, $(M_{1})$, $(M_{2})$, $(f_{1})$, $(f_{2})$ and $(f_{6})$ hold, then there exists a critical point of $I$ corresponding to negative critical value for any $\lambda\in(0,\lambda_{1})$.

Proof. By Lemma 3.1, for any $\lambda\in(0,\lambda_{1})$, we can see that there exits a local minimizer of $I$ in $B_{\varrho}$. The following proof shows this minimizer is not zero. By $(f_{1})$, there exists $\varsigma_{1}>0$ such that

[TABLE]

for all $x\in\Upsilon_{\varsigma_{1}}(\bar{x})$ and $t\in\mathbb{R}$. Choosing $\varphi_{1}\in C_{0}^{\infty}(\Upsilon_{\varsigma_{1}}(\bar{x}),\mathbb{R})\setminus\{0\}$, by $(M_{2})$, $(f_{2})$, (18) and $(f_{6})$, there exists $M_{4}>0$ such that

[TABLE]

for $\theta>0$ small enough. Hence

[TABLE]

Similar to the proof of Theorem 3.5 in [8], there exists $u^{**}\in B_{\varrho}\setminus\partial B_{\varrho}$ such that

[TABLE]

The proof of this lemma is finished.

Proof of Theorem 1.2. From Lemmas 3.1-3.4, we can see that problem (1) possesses at least two solutions for any $\lambda\in(0,\lambda_{1})$. $\Box$

4 Proof of Theorem 1.3

In this section, we use Lemma 1.1 to obtain infinitely many critical points of $I$. The following lemmas will show that $I$ satisfies the conditions of Lemma 1.1.

Lemma 4.1**.**

Suppose (6), $(V)$, $(M_{1})$, $(M_{2})$, $(f_{2})$ and $(f_{3})$ hold, then I satisfies $(C_{1})$.

Proof. Let $\{v_{j}\}_{j=1}^{\infty}$ be a completely orthogonal basis of $E$ and $X_{k}=\bigoplus_{j=1}^{k}S_{j}$, where $S_{j}=span\{v_{j}\}$. For any $q\in[2,2^{*})$, we set

[TABLE]

It follows from Lemma 2.10 in [18] that $\mathcal{P}_{k}(q)\rightarrow 0$ as $k\rightarrow\infty$ for any $q\in[2,2^{*})$. By $(f_{3})$, there exists $\rho_{1}\in(0,1)$ such that

[TABLE]

for all $|t|\leq\rho_{1}$ and $x\in\mathbb{R}^{N}$ . Set

[TABLE]

Then there exists $k_{0}>0$ such that $\mathcal{H}_{k}\leq\frac{1}{2}\min\left\{\frac{1}{2},\frac{m_{0}}{2}+\sigma_{1}\right\}\rho_{1}$ for all $k\geq k_{0}$. Then for any $u\in X_{k_{0}}^{\perp}\bigcap\partial B_{\rho_{1}}$, it follows from $(M_{1})$, $(M_{2})$, $(f_{2})$, (19) and (20) that

[TABLE]

We finish the proof of this lemma.

Lemma 4.2**.**

Suppose (6), $(V)$, $(M_{2})$, $(f_{2})$ and $(f_{4})$ hold, then $I$ satisfies $(C_{2})$.

Proof. Set $\tilde{X}_{m}=\bigoplus_{j=1}^{m}S_{j}$, where $S_{j}$ is defined in Lemma 4.1. For any $u\in\tilde{X}_{m}\setminus\{0\}$ and $\vartheta>0$, set

[TABLE]

Similar to [29], there exists $\vartheta_{0}>0$ such that

[TABLE]

for all $u\in\tilde{X}_{m}\setminus\{0\}$. Then there exists $\kappa>0$ such that

[TABLE]

for all $u\in\tilde{X}_{m}\setminus\{0\}$, where $\Lambda_{\vartheta_{0}}(u)=\Gamma_{\vartheta_{0}}(u)\bigcap\Upsilon_{\kappa}(0)$. It follows from $(f_{4})$ that there exists $L_{2}>0$ such that

[TABLE]

for all $u\in\tilde{X}_{m}$ and $x\in\Lambda_{\vartheta_{0}}(u)$ with $\|u\|_{E}\geq L_{2}$. We can choose $\varsigma_{m}>L_{2}$, then for any $u\in\tilde{X}_{m}\setminus B_{\varsigma_{m}}$, it follows from (7), $(M_{2})$, (9), (14), (22) and (23) that

[TABLE]

Then there exists $r_{m}>\xi$ such that $I(u_{m})\leq 0$ for all $u\in\tilde{X}_{m}\setminus B_{r_{m}}$, which proves this lemma.

Lemma 4.3**.**

Suppose the conditions of Theorem 1.3 hold, then $I$ satisfies the $(PS)^{*}$ condition.

Proof. The proof is similar to Lemma 3.3, we omit it here.

Proof of Theorem 1.3. By Lemmas 4.1-4.3 and Lemma 1.1, $I$ possesses infinitely many distinct critical points corresponding to positive critical values. $\Box$

5 Acknowledgments

The authors are grateful to the referees for the helpful comments which improve the writing of the paper. This paper was finished when D.-L. Wu was visiting Utah State University with the support of China Scholarship Council(No.201708515186); he is grateful to the members in the Department of Mathematics and Statistics at Utah State University for their invitation and hospitality.