Multiple Solutions for Nonlocal Fourth-Order Equation with Concave–Convex Nonlinearities

Abstract

This paper is devoted to a class of general nonlocal fourth-order elliptic equation with concave–convex nonlinearities. First, using the $Z_2$-mountain pass theorem in critical point theory, we obtain the existence of infinitely many large energy solutions. Then, using the dual fountain theorem, we prove that the equation has infinitely many negative energy solutions, whose energy converges at 0. Our results extend and complement existing findings in the literature.

1. Introduction and Main Results

In this paper, we are concerned with the existence of solutions for the following biharmonic equations of Kirchhoff type:

$${\mathsf{\Delta }}^{2}u-(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^{2}dx)\mathsf{\Delta }u+V\left(x\right)u=f(x,u)+\mu g(x,u),x\in {\mathbb{R}}^N,$$

(1)

where ${\mathsf{\Delta }}^2=\mathsf{\Delta }\left(\mathsf{\Delta }\right)$ is the biharmonic operator; $a>0,b\ge 0$ are constants, $N\ge 5$; and $\mu$ is a parameter, $V\in C({\mathbb{R}}^N,\mathbb{R})$ and $f,g\in C({\mathbb{R}}^N\times \mathbb{R},\mathbb{R})$. The presence of the integral term $({\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx)\mathsf{\Delta }u$ implies the problem (1) is no longer a pointwise identity, thus, (1) is often called nonlocal. The nonlinearity $f(x,u)+\mu g(x,u)$ may involve a combination of concave–convex terms, making it challenging to verify the mountain pass geometry of the associated energy functional, obtain a (PS) sequence, and show the compactness of the (PS) sequence. In recent years, there has been increased attention the equations with concave–convex terms; refer to [1,2,3] and the references therein.

For $a=1,b=0$, (1) becomes a fourth-order equation

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

(2)

In recent years, the existence and multiplicity of solutions for (2) (such as positive, negative, sign-changing, and high-energy solutions) have been the subject of extensive mathematical studies; see [4,5] and the references therein. In particular, for $h(x,u)=K\left(x\right)f\left(u\right)+{\xi \left(x\right)|u|}^{q-2}u$ and $1<q<2$, by working in weighted Sobolev spaces and using a variational method, [3] proved that the equation had two nontrivial solutions. Moreover, when the equation involves p-Laplacian, [6] showed the existence of infinitely many nontrivial solutions with small energy using various variational methods.

Without ${\mathsf{\Delta }}^{2}u$, (1) is the Kirchhoff equation

$$-(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^{2}dx)\mathsf{\Delta }u+V\left(x\right)u=h(x,u),x\in {\mathbb{R}}^N.$$

(3)

By using Nehari manifold and fibering map methods, ref. [7] considered problem (3) with concave and convex nonlinearities, and the existence of multiple positive solutions were researched. By using the monotonicity trick and a new version of the global compactness lemma, ref. [8] considered (3) with superlinear nonlinearities and obtained the existence of positive ground state solutions. For other important results, see [9,10,11,12] and the references therein.

When $a>0,b>0$, problem (1) may be seen as a linear couple of (2) and (3). By using the variational methods and the Nehari method, [13] researched ground state solutions and lower energy sign-changing solutions for (1). As $f(x,u)$ satisfies Ambrosetti–Rabinowitz-type conditions, ref. [14] studied the existence of infinitely many solutions for (1) using the symmetric mountain pass theorem. When (1) involves concave–convex nonlinearities, ref. [15] proved that there were at least two positive solutions, one of which was a positive ground state solution, using the Nehari manifold, Ekeland variational principle, and the theory of Lagrange multipliers. However, very little work has researched infinitely many high-energy solutions and infinitely many negative energy solutions for problem (1).

Ref. [2] established the existence of infinitely many high-energy solutions using fountain theorem, and assumed that

$\left(V\right)$ $V\in C({\mathbb{R}}^3,\mathbb{R})$ and ${inf}_{x\in {\mathbb{R}}^3}V\left(x\right)>0$, and for any $M>0$, one has

$$meas\left(\{x\in {\mathbb{R}}^3:V\left(x\right)\le M\}\right)<+\infty .$$

$\left(g_1^{\prime }\right)$ There exist $1<q_1<q_2<2$ and functions $h_i\left(x\right)\in L^{{\displaystyle \frac{2}{2-q_i}}}({\mathbb{R}}^3,{\mathbb{R}}^{+})(i=1,2)$, such that

$$|g(x,u)|\le h_1{\left(x\right)|u|}^{q_1-1}+h_2\left(x\right){|u|}^{q_2-1},\forall (x,u)\in {\mathbb{R}}^3\times \mathbb{R}.$$

$\left(g_2\right)$ There has $\nu \in (1,2)$, such that $0<ug(x,u)\le \nu G(x,u)$, $\forall (x,u)\in {\mathbb{R}}^3\times \mathbb{R}$.

$\left(g_3\right)$ $g(x,-u)=-g(x,u),\forall (x,u)\in {\mathbb{R}}^3\times \mathbb{R}$;

$\left(F_1\right)$ $f\in C({\mathbb{R}}^3,\mathbb{R})$, and there exist $c_1>0$ and $p\in (2,6)$, such that

$$|f(x,u)|\le c_1{(1+|u|}^{p-1}),\forall (x,u)\in {\mathbb{R}}^3\times \mathbb{R}.$$

$\left(F_2\right)$ ${lim}_{|u|\to 0}{\displaystyle \frac{f(x,u)}{u}}=0$ uniformly in $x\in {\mathbb{R}}^3$.

$\left(F_3\right)$ There exists $\theta >4$, such that

$$0<\theta F(x,u)\le uf(x,u),\forall (x,u)\in {\mathbb{R}}^3\times \mathbb{R}.$$

$\left(F_4\right)$ $f(x,-u)=-f(x,u)$, $\forall (x,u)\in {\mathbb{R}}^3\times \mathbb{R}$.

In our work, inspired by the above-mentioned papers, we will focus on infinitely many solutions for (1) involving concave–convex nonlinearities. To state our results, we impose the following conditions on V, f, and g:

$\left(V_1\right)$ $V\in C({\mathbb{R}}^N,\mathbb{R})$ and ${inf}_{x\in {\mathbb{R}}^N}V\left(x\right)>0$, and there exists a constant $d_0>0$, such that

$$\underset{|y|\to \infty }{lim}meas\left(\right\{x\in {\mathbb{R}}^N:|x-y|\le d_0,V\left(x\right)\le M\left\}\right)=0,\forall M>0,$$

where $meas$ denotes the Lebesgue measure on ${\mathbb{R}}^N$.

$\left(g_1\right)$ There exist constants $1<q_1<q_2<2$, such that

$$|g(x,u)|\le h_1{\left(x\right)|u|}^{q_1-1}+h_2\left(x\right){|u|}^{q_2-1},\forall (x,u)\in {\mathbb{R}}^N\times \mathbb{R},$$

where functions $h_i\left(x\right)\in L^{{\beta }_i}({\mathbb{R}}^N,{\mathbb{R}}^{+})$, ${\beta }_i\in ({\displaystyle \frac{2^{\ast \ast }}{2^{\ast \ast }-q_i}},{\displaystyle \frac{2}{2-q_i}}],i=1,2$, $2^{\ast \ast }={\displaystyle \frac{2N}{N-4}}$ is the critical Sobolev exponent.

$\left(g_2^{\prime }\right)$ There exists $\nu \in (1,2)$, such that $0<ug(x,u)\le \nu G(x,u)$, $\forall (x,u)\in {\mathbb{R}}^N\times \mathbb{R}$.

$\left(f_1\right)$ $f\in C({\mathbb{R}}^N\times \mathbb{R},\mathbb{R})$, and there exist $c_0>0$ and $p\in (2,2^{\ast \ast })$, such that

$$|f(x,u)|\le c_0(|u|+{|u|}^{p-1}),\forall (x,u)\in {\mathbb{R}}^N\times \mathbb{R}.$$

$\left(f_2\right)$ ${lim}_{|u|\to \infty }{\displaystyle \frac{|F(x,u)|}{{|u|}^4}}=+\infty$, i.e., $x\in {\mathbb{R}}^N$ and

$$F(x,u)\ge 0,\forall (x,u)\in {\mathbb{R}}^N\times \mathbb{R}.$$

$\left(f_3\right)$ There exist $L_0>0,\theta \ge 4$ and $c_1>0$, such that

$$uf(x,u)-\theta F(x,u)+c_1{|u|}^2\ge 0,\forall (x,|u|)\in {\mathbb{R}}^N\times [L_0,+\infty ).$$

$\left(f_3^{\prime }\right)$ There exists $\theta >4$, such that

$$0<\theta F(x,u)\le uf(x,u),\forall (x,u)\in {\mathbb{R}}^N\times \mathbb{R}.$$

$\left(f_4\right)$ $f(x,-u)=-f(x,u)$, $\forall (x,u)\in {\mathbb{R}}^N\times \mathbb{R}$.

$\left(h\right)$ $h:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, and $h\in L^{{\displaystyle \frac{2}{2-r}}}\left({\mathbb{R}}^N\right)(1<r<2)$.

Now, we are ready to state the main results of this paper.

Theorem 1. 

Suppose that $\left(V_1\right)$, $\left(g_1\right),\left(g_2\right)$, $\left(f_1\right)-\left(f_4\right)$ are satisfied. Then, there exists ${\mu }_0>0$, such that the problem (1) has infinitely many large energy solutions for $|\mu |\le {\mu }_0$.

Theorem 2. 

When $g(x,u)={h\left(x\right)|u|}^{r-2}u,1<r<2$. Suppose that $\left(V_1\right)$, $\left(h\right)$, $\left(f_1\right),\left(f_3^{\prime }\right),\left(f_4\right)$ are satisfied. Then, for every $\mu >0$, the problem (1) has a sequence of solutions $\left\{u_k\right\}$ with $J\left(u_k\right)\to 0$ as $k\to \infty$.

Remark 1. 

It is easy to know that $\left(f_1\right)$ is weaker than the combination of $\left(F_1\right)$ and $\left(F_2\right)$, and the combination $\left(f_2\right)$ and $\left(f_3\right)$ is much weaker than $\left(F_3\right)$. We remove the condition $\left(g_3\right)$ completely.

2. Preliminaries and Functional Setting

In this paper, we make use of the following notations: the $L^r$-norm ($1\le r\le +\infty$) by ${|·|}_r$. C denotes various positive constants, which may vary from line to line. Let

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

with the inner product and norm

$${(u,v)}_H={\int }_{{\mathbb{R}}^N}(\mathsf{\Delta }u\mathsf{\Delta }v+\nabla u\nabla v+uv)dx,{\parallel u\parallel }_H={(u,u)}_H^{{\displaystyle \frac{1}{2}}}.$$

In view of the potential $V\left(x\right)$, we consider our working space

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

Thus, E is a Hilbert space with the inner product and norm

$$(u,v)={\int }_{{\mathbb{R}}^N}(\mathsf{\Delta }u\mathsf{\Delta }v+a\nabla u\nabla v+V\left(x\right)uv)dx,\parallel u\parallel ={(u,u)}^{{\displaystyle \frac{1}{2}}}.$$

In the rest of this article, we use the norm $\parallel ·\parallel$ in E. Obviously, E is continuously embedded in $L^p\left({\mathbb{R}}^N\right)$ for $p\in [2,2^{\ast \ast }]$, then for any $p\in [2,2^{\ast \ast }]$, there exists $S_p>0$, such that

$${|u|}_p\le S_p\parallel u\parallel ,\forall u\in E.$$

(4)

In order to obtain the main results, we need the following lemmas and theorems. Motivated by Lemma 3.4 in [16], one can prove Lemma 1 below in the same way. Here, we omit it.

Lemma 1. 

Under the assumption $\left(V_1\right)$, then the embedding $E\hookrightarrow L^p\left({\mathbb{R}}^N\right)$ is compact for $p\in [2,2^{\ast \ast })$.

Next, we define the energy functional J on E by

$$J\left(u\right)={\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{b}{4}}({\int }_{{\mathbb{R}}^N}{|\nabla u|}^2{dx)}^2-\mu {\int }_{{\mathbb{R}}^N}G(x,u)dx-{\int }_{{\mathbb{R}}^N}F(x,u)dx,\forall u\in E.$$

(5)

It is not difficult to prove that the functional J is of class $C^1$ in E, and that

$$\begin{array}{ccc}\hfill \langle J^{\prime }\left(u\right),v\rangle & =& {\displaystyle {\int }_{{\mathbb{R}}^N}[\mathsf{\Delta }u·\mathsf{\Delta }v+a\nabla u\nabla v+V\left(x\right)uv+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^{2}dx·\nabla u\nabla v}\hfill \\ \hfill & & -\mu g(x,u)v-f(x,u)v]dx\hfill \end{array}$$

(6)

for all $u,v\in E$. Obviously, it can be proven that if u is a critical point of J, then it is a weak solution for the problem (1).

Recall that a sequence $\left\{u_n\right\}\subset E$ is said to be a Palais–Smale sequence at the level $c\in \mathbb{R}$ ((PS)_c-sequence for short) if $J\left(u_n\right)\to c$ and $J^{\prime }\left(u_n\right)\to 0$ as $n\to \infty$. J is said to satisfy the (PS)_c condition and that the (PS)_c-sequence has a convergent subsequence.

To obtain our first theorem, the following theorem (see [17], Theorem 9.12) is used.

Theorem 3 

($Z_2$-Mountain Pass Theorem). Let X be an infinite dimensional Banach space, $X=Y\oplus Z$, where Y is a finite dimensional space. If $J\in C^1(X,\mathbb{R})$ satisfies

(1) 

$J\left(0\right)=0,J(-u)=-J\left(u\right)$ for all $u\in X$;

(2) 

there exist constants $\rho ,\alpha >0$, such that ${J|}_{\partial B_{\rho }\cap Z}\ge \alpha$;

(3) 

for any finite dimensional subspace $\tilde{X}\subset X$, there is $R=R\left(\tilde{X}\right)>0$ such that $J\left(u\right)\le 0$ on $\tilde{X}\setminus B_R$;

(4) 

the Palais–Smale condition holds above 0, i.e., any sequence $\left\{u_n\right\}$ in X that satisfies $J\left(u_n\right)\to c>0$ and $J^{\prime }\left(u_n\right)\to 0$ contains a convergent subsequence.

Then, J possesses an unbounded sequence of critical values.

Definition 1 

(see [18]). Let X be a Banach space, $I\in C^1(X,\mathbb{R})$, and $c\in \mathbb{R}.$ The function I satisfies the $(P$S${)}_c^{\ast }$ condition [with respect to $\left(Y_n\right)$], if any sequence $\left\{u_{n_j}\right\}\subset X$, such that

$$u_{n_j}\in Y_{n_j},I\left(u_{n_j}\right)\to c,I{\mid }_{Y_{n_j}}^{\prime }\left(u_{n_j}\right)\to 0,n_j\to \infty$$

has a convergent subsequence.

Theorem 4 

(see [19], dual fountain theorem). Let $I\in C^1(X,\mathbb{R})$ satisfy $I(-u)=I\left(u\right).$ Assume that, for every $k>k_0$, there exists ${\rho }_k>{\gamma }_k>0$, such that

$\left(B_1\right)$ $a_k:={inf}_{u\in Z_k,\parallel u\parallel ={\rho }_k}I\left(u\right)\ge 0$,

$\left(B_2\right)$$b_k:={max}_{u\in Y_k,\parallel u\parallel ={\gamma }_k}I\left(u\right)<0,$

$\left(B_3\right)$$d_k:={inf}_{u\in Z_k,\parallel u\parallel \le {\rho }_k}I\left(u\right)\to 0ask\to +\infty ,$

$\left(B_4\right)$ If I satisfies the ${\left(PS\right)}_c^{\ast }$ condition for every $c\in [d_{k_0},0)$,

  then I has a sequence of negative critical values converging to 0.

  Let $\left\{e_j\right\}$ be a total orthonormal basis of E and define $X_j={\mathbb{R}}_{e_j}$,

$$Y_k=\underset{j=1}{\overset{k}{⨁}}{X}_j,Z_k=\underset{j=k+1}{\overset{\infty }{⨁}}{X}_j,\forall k\in \mathbb{Z}.$$

Lemma 2. 

Suppose that $\left(V_1\right)$ is satisfied. Then, for $2\le p<2^{\ast \ast }$, we have

$${\beta }_p\left(k\right):=\underset{u\in Z_k,\parallel u\parallel =1}{sup}{|u|}_p\to 0,k\to \infty .$$

Proof. 

It is clear that $0<{\beta }_{k+1}<{\beta }_k$, so that ${\beta }_k\to \beta \ge 0$, as $k\to \infty$. For every $k\in \mathbb{N}$, there exists $u_k\in Z_k$, such that $|u_k{|}_2>{\displaystyle \frac{{\beta }_k}{2}}$ and $\parallel u_k\parallel =1$. For any $v\in E$, writing $v={\sum }_{j=1}^{\infty }{c}_j{e}_j$, using the Canchy–Schwartz inequality, one has

$$\begin{array}{ccc}\hfill |(u_k,v)|& =& |(u_k,\sum _{j=1}^{\infty }{c}_j{e}_j)|=|(u_k,\sum _{j=k+1}^{\infty }{c}_j{e}_j)|\hfill \\ \hfill & \le & \parallel u_k\parallel \parallel \sum _{j=k+1}^{\infty }{c}_j{e}_j\parallel ={\left(\sum _{j=k+1}^{\infty }{c}_j^2\right)}^{1/2}\to 0,\hfill \end{array}$$

as $k\to \infty$, which implies that $u_k\rightharpoonup 0$. By Lemma 1, the compact embedding of $E\hookrightarrow L^p\left({\mathbb{R}}^N\right)$ $(2\le p<2^{\ast \ast })$ implies that $u_k\to 0$ in $L^p\left({\mathbb{R}}^N\right)$. Hence, letting $k\to \infty$, we obtain $\beta =0$, which completes the proof. □

3. Proof of Theorem 1

Evidently, J is even by $\left(f_4\right)$, $J\left(0\right)=0$. Next, we will prove that $J\left(u\right)$ satisfies (2)–(4) of Theorem 3. Specifically, Lemma 3 proves that J satisfies (2); Lemma 4 proves that J satisfies (3); Lemma 5 and Lemma 6 prove that J satisfies (4).

By Lemma 2, we can choose an integer $m\ge 1$, such that

$${|u|}_2^2\le {\displaystyle \frac{1}{2c_0}}{\parallel u\parallel }^2,\forall u\in Z_m,$$

(7)

where $c_0$ can be found in $\left(f_1\right)$. Let $V=Y_{m-1},X=Z_m$, then $E=V\oplus X$ and V is finite dimensional.

Lemma 3. 

Under assumptions $\left(V_1\right),\left(g_1\right)$ and $\left(f_1\right)$, then there exist ${\mu }_0>0$, $\rho ,\alpha >0$, such that $J\left(u\right)\ge \alpha$ for $\parallel u\parallel =\rho$ and $|\mu |\le {\mu }_0$.

Proof. 

By $\left(f_1\right)$ and $F(x,z)={\int }_0^{1}f(x,tz)zdt$, one obtains that

$${\displaystyle F(x,z)\le \frac{c_0}{2}{|z|}^2+\frac{c_0}{p}{|z|}^p,\forall (x,z)\in ({\mathbb{R}}^N,\mathbb{R}).}$$

Thus, by (5), (11), $q_1<q_2$, (7), and (4), for any $u\in E$ satisfying $\parallel u\parallel \le 1$, we have

$$\begin{array}{ccc}\hfill J\left(u\right)& \ge & {\displaystyle \frac{1}{2}{\parallel u\parallel }^2-|\mu |{\int }_{{\mathbb{R}}^N}|G(x,u)|dx-{\int }_{{\mathbb{R}}^N}F(x,u)dx}\hfill \\ \hfill & \ge & {\displaystyle \frac{1}{2}{\parallel u\parallel }^2-|\mu |\left(\frac{1}{q_1}|h_1{|}_{{\beta }_1}{S}_{\frac{q_1{\beta }_1}{{\beta }_1-1}}^{q_1}{\parallel u\parallel }^{q_1}+\frac{1}{q_2}|h_2{|}_{{\beta }_2}{S}_{\frac{q_2{\beta }_2}{{\beta }_2-1}}^{q_2}{\parallel u\parallel }^{q_2}\right)-\frac{c_0}{2}{|u|}_2^2-\frac{c_0}{p}{|u|}_p^p}\hfill \\ \hfill & \ge & {\displaystyle \frac{1}{2}{\parallel u\parallel }^2-|\mu |\left(\frac{1}{q_1}|h_1{|}_{{\beta }_1}{S}_{\frac{q_1{\beta }_1}{{\beta }_1-1}}^{q_1}+\frac{1}{q_2}{|h_2|}_{{\beta }_2}{S}_{\frac{q_2{\beta }_2}{{\beta }_2-1}}^{q_2}\right){\parallel u\parallel }^{q_1}-\frac{1}{4}{\parallel u\parallel }^2-\frac{c_0{S}_p^p}{p}{\parallel u\parallel }^p}\hfill \\ \hfill & =& {\parallel u\parallel }^{q_1}\left[{\displaystyle \frac{1}{4}}{\parallel u\parallel }^{2-q_1}-{\displaystyle \frac{c_0{S}_p^p}{p}}{\parallel u\parallel }^{p-q_1}-|\mu |M\right],\hfill \end{array}$$

where $M:={\displaystyle \frac{1}{q_1}}|h_1{|}_{{\beta }_1}{S}_{{\displaystyle \frac{q_1{\beta }_1}{{\beta }_1-1}}}^{q_1}+{\displaystyle \frac{1}{q_2}}{|h_2|}_{{\beta }_2}{S}_{{\displaystyle \frac{q_2{\beta }_2}{{\beta }_2-1}}}^{q_2}$. Set

$$g\left(t\right)={\displaystyle \frac{1}{4}}{t}^{2-q_1}-{\displaystyle \frac{c_0{S}_p^p}{p}}{t}^{p-q_1}=t^{2-q_1}\left[{\displaystyle \frac{1}{4}}-{\displaystyle \frac{c_0{S}_p^p}{p}}{t}^{p-2}\right],t\ge 0.$$

It is easy to know that there exists a constant $0<t_0\le 1$ satisfying $g\left(t\right)>0$ for any $t\in (0,t_0]$. Choosing $\parallel u\parallel =t_0:=\rho$, we obtain that, for any $\mu$ satisfying $|\mu |\le {\displaystyle \frac{g\left(\rho \right)}{2M}}:={\mu }_0$ and $\parallel u\parallel =\rho$, then

$$J\left(u\right)\ge {\rho }^{q_1}[g\left(\rho \right)-|\mu |M]\ge {\displaystyle \frac{{\rho }^{q_1}g\left(\rho \right)}{2}}:=\alpha .$$

Thus, we finish the proof. □

Lemma 4. 

Under $\left(V_1\right),\left(g_1\right)$ and $\left(f_2\right)$, for every finite dimensional subspace $Y\subset E$, there exists $R=R\left(Y\right)$, such that $J\left(u\right)\le 0$ if $\parallel u\parallel \ge R$.

Proof. 

Arguing indirectly, we assume that there exists a sequence $\left\{u_n\right\}\subset Y$, such that $\parallel u_n\parallel \to \infty$ and $J\left(u_n\right)\ge 0$ for any $n\in \mathbb{N}$. Let $w_n={\displaystyle \frac{u_n}{\parallel u_n\parallel }}$, going if necessary to a subsequence, we may assume that $w_n\left(x\right)\to w\left(x\right)$ for a.e. $x\in {\mathbb{R}}^N$. Since Y is finite dimensional, then $w_n\to w$ in Y, and so $\parallel w\parallel =1$. Set

$$A:=\{x\in {\mathbb{R}}^N:v\left(x\right)\ne 0\},$$

thus $meas\left(A\right)>0$ and for a.e. $x\in A$, we have ${lim}_{n\to \infty }|u_n\left(x\right)|=+\infty$. Obviously,

$${\displaystyle \frac{({\int }_{{\mathbb{R}}^N}|\nabla u_n{{|}^{2}dx)}^2}{\parallel u_n{\parallel }^4}}\le {\displaystyle \frac{1}{a^2}}.$$

(8)

From $\left(f_2\right)$ and Fatou’s lemma, one has

$$\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{{\int }_{{\mathbb{R}}^N}F(x,u_n)dx}{\parallel u_n{\parallel }^4}}\ge {\int }_A\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{F(x,u_n)w_n^2\left(x\right)}{|u_n{\left(x\right)|}^4}}dx=+\infty .$$

(9)

By the equality $G(x,z)={\int }_0^{1}g(x,tz)zdt$, $\left(g_1\right)$, the Holder inequality and (4), we have that

$$\begin{array}{ccc}\hfill {\displaystyle {\int }_{{\mathbb{R}}^N}|G(x,u)|dx}& \le & {\displaystyle {\int }_{{\mathbb{R}}^N}\left(\frac{1}{q_1}|h_1{\left(x\right)||u|}^{q_1}+\frac{1}{q_2}|h_2{\left(x\right)||u|}^{q_2}\right)dx}\hfill \\ \hfill & \le & {\displaystyle \frac{1}{q_1}|h_1{|}_{{\beta }_1}{|u|}_{\frac{q_1{\beta }_1}{{\beta }_1-1}}^{q_1}+\frac{1}{q_2}|h_2{|}_{{\beta }_2}{|u|}_{\frac{q_2{\beta }_2}{{\beta }_2-1}}^{q_2}}\hfill \\ \hfill & \le & {\displaystyle \frac{1}{q_1}|h_1{|}_{{\beta }_1}{S}_{\frac{q_1{\beta }_1}{{\beta }_1-1}}^{q_1}{\parallel u\parallel }^{q_1}+\frac{1}{q_2}|h_2{|}_{{\beta }_2}{S}_{\frac{q_2{\beta }_2}{{\beta }_2-1}}^{q_2}{\parallel u\parallel }^{q_2}.}\hfill \end{array}$$

(10)

In view of $1<q_1<q_2<2$ and (9), one has

$${\displaystyle \frac{{\int }_{{\mathbb{R}}^N}G(x,u_n)dx}{\parallel u_n{\parallel }^4}}\le {\displaystyle \frac{1}{q_1}}|h_1{|}_{{\beta }_1}{S}_{{\displaystyle \frac{q_1{\beta }_1}{{\beta }_1-1}}}^{q_1}{\parallel u\parallel }^{q_1-4}+{\displaystyle \frac{1}{q_2}}|h_2{|}_{{\beta }_2}{S}_{{\displaystyle \frac{q_2{\beta }_2}{{\beta }_2-1}}}^{q_2}{\parallel u\parallel }^{q_2-4}\to 0,$$

(11)

as $n\to \infty$. Thus, combining (5), (8), (11) with $J\left(u_n\right)\ge 0$, we obtain that

$$\begin{array}{ccc}\hfill & & {\displaystyle \underset{n\to \infty }{lim\; inf}\frac{{\int }_{{\mathbb{R}}^N}F(x,u_n)dx}{\parallel u_n{\parallel }^4}\le \underset{n\to \infty }{lim\; sup}\frac{{\int }_{{\mathbb{R}}^N}F(x,u_n)dx}{\parallel u_n{\parallel }^4}}\hfill \\ \hfill & & {\displaystyle \le \underset{n\to \infty }{lim\; sup}\left[\frac{1}{2\parallel u_n{\parallel }^2}+\frac{b({\int }_{{\mathbb{R}}^N}|\nabla u_n{{|}^{2}dx)}^2}{4\parallel u_n{\parallel }^4}\right]-\underset{n\to \infty }{lim}\frac{\mu {\int }_{{\mathbb{R}}^N}G(x,u_n)dx}{\parallel u_n{\parallel }^4}}\hfill \\ \hfill & & {\displaystyle -\underset{n\to \infty }{lim\; inf}\frac{J\left(u_n\right)}{\parallel u_n{\parallel }^4}\le \frac{b}{4a^2},}\hfill \end{array}$$

which contradicts with (9). Thus, we obtain the conclusion. □

Lemma 5. 

Under assumptions $\left(V_1\right),\left(g_1\right)$ and $\left(f_1\right),\left(f_2\right),\left(f_3\right)$, any Palais–Smale sequence for the functional J is bounded in E.

Proof. 

We suppose that $\left\{u_n\right\}$ is a Palais–Smale sequence of J, that is for some $c\in \mathbb{R}$, $J\left(u_n\right)\to c,J^{\prime }\left(u_n\right)\to 0$, as $n\to \infty$. We need to prove that $\left\{u_n\right\}$ is bounded in E. Otherwise, up to subsequence, one can assume that $\parallel u_n\parallel \to \infty$, as $n\to \infty$. Let $w_n={\displaystyle \frac{u_n}{\parallel u_n\parallel }}$, going if necessary to a subsequence, we assume that $w_n\to w$ in $L^s\left({\mathbb{R}}^N\right)(2\le s<2^{\ast \ast })$, $w_n\left(x\right)\to w\left(x\right)$ for a.e.$x\in {\mathbb{R}}^N$. Set $\mathsf{\Omega }=\{x\in {\mathbb{R}}^N:w\left(x\right)\ne 0\}$. If meas$\left(\mathsf{\Omega }\right)>0$, similar to the proof of Lemma 3, we can obtain a contradiction. Hence, meas$\left(\mathsf{\Omega }\right)=0$, which means $w\left(x\right)=0$ for a.e. $x\in {\mathbb{R}}^N$ and $w_n\to 0$ in $L^s\left({\mathbb{R}}^N\right)(2\le s<2^{\ast \ast })$.

By $\left(f_1\right)$ and $p>2$, we obtain that

$$|f(x,z)z|\le c_0{(|z|}^2+{|z|}^p)\le 2c_0{|z|}^2,\forall (x,|z|)\in {\mathbb{R}}^N\times [0,1].$$

Again, by $\left(f_1\right)$, there exists $M>0$, such that

$$|f(x,z)z|\le c_0{(|z|}^2+{|z|}^p{)\le M\le M|z|}^2,\forall (x,|z|)\in {\mathbb{R}}^N\times [1,L_0].$$

Hence, from above two inequalities we have

$$|f(x,z)z|\le (M+2c_0){|z|}^2,\forall (x,|z|)\in {\mathbb{R}}^N\times [0,L_0].$$

(12)

Using the equality $F(x,z)={\int }_0^{1}f(x,tz)zdt$, we obtain that

$$|F(x,z)|\le {\displaystyle \frac{1}{2}}(M+2c_0){|z|}^2,\forall (x,z)\in {\mathbb{R}}^N\times [0,L_0].$$

(13)

Combining (12) with (13), one has

$$|zf(x,z)-\theta F(x,z)|\le \widetilde{c_1}{|z|}^2,\forall (x,|z|)\in {\mathbb{R}}^N\times [0,L_0],$$

where $\widetilde{c_1}=(\theta /2+1)(M+2c_0)$. Then, by $\left(f_3\right)$, we deduce that

$$zf(x,z)-\theta F(x,z)\ge -(c_1+\widetilde{c_1}){|z|}^2,\forall (x,z)\in {\mathbb{R}}^N\times \mathbb{R}.$$

(14)

By the equality $G(x,z)={\int }_0^{1}g(x,tz)zdt$, $\left(g_1\right)$ and the Hölder inequality, one obtains

$$\begin{array}{ccc}\hfill & & {\displaystyle |{\int }_{{\mathbb{R}}^N}(g(x,u)u-\theta G(x,u))dx|\le {\int }_{{\mathbb{R}}^N}|g(x,u)u|+\theta |G(x,u)|dx}\hfill \\ \hfill & & {\displaystyle \le {\int }_{{\mathbb{R}}^N}(|h_1{\left(x\right)||u|}^{q_1}+|h_2{\left(x\right)||u|}^{q_2})dx+{\int }_{{\mathbb{R}}^3}(\frac{\theta }{q_1}|h_1{\left(x\right)||u|}^{q_1}+\frac{\theta }{q_2}|h_2{\left(x\right)||u|}^{q_2})dx}\hfill \\ \hfill & & \le (1+{\displaystyle \frac{\theta }{q_1}})|h_1{|}_{{\beta }_1}{|u|}_{{\displaystyle \frac{q_1{\beta }_1}{{\beta }_1-1}}}^{q_1}+(1+{\displaystyle \frac{\theta }{q_2}})|h_2{|}_{{\beta }_2}{|u|}_{{\displaystyle \frac{q_2{\beta }_2}{{\beta }_2-1}}}^{q_2}\hfill \\ \hfill & & {\displaystyle \le (1+\frac{\theta }{q_1})|h_1{|}_{{\beta }_1}{S}_{\frac{q_1{\beta }_1}{{\beta }_1-1}}^{q_1}{\parallel u\parallel }^{q_1}+(1+\frac{\theta }{q_2})|h_2{|}_{{\beta }_2}{S}_{\frac{q_2{\beta }_2}{{\beta }_2-1}}^{q_2}{\parallel u\parallel }^{q_2}.}\hfill \end{array}$$

For $\parallel u_n\parallel \to \infty$ and $1<q_1<q_2<2$, we have that

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{|{\int }_{{\mathbb{R}}^3}(g(x,u)u-\theta G(x,u))dx|}{\parallel u_n{\parallel }^2}}\hfill \\ \hfill & & {\displaystyle \le (1+\frac{\theta }{q_1})|h_1{|}_{{\beta }_1}{S}_{\frac{q_1{\beta }_1}{{\beta }_1-1}}^{q_1}{\parallel u\parallel }^{q_1-2}+(1+\frac{\theta }{q_2})|h_2{|}_{{\beta }_2}{S}_{\frac{q_2{\beta }_2}{{\beta }_2-1}}^{q_2}{\parallel u\parallel }^{q_2-2}\to 0,}\hfill \end{array}$$

as $n\to \infty$.

Hence, it follows from (5), (6), $\theta \ge 4$, and (14) that

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{|{\int }_{{\mathbb{R}}^3}\theta J\left(u_n\right)-\langle J^{\prime }\left(u_n\right),u_n\rangle |}{\parallel u_n{\parallel }^2}}\hfill \\ \hfill & & {\displaystyle =(\frac{\theta }{2}-1)\parallel u_n{\parallel }^2+\frac{1}{\parallel u_n{\parallel }^2}(\frac{\theta }{4}-1)b({\int }_{{\mathbb{R}}^N}|\nabla u_n{{|}^{2}dx)}^2+\frac{\mu }{\parallel u_n{\parallel }^2}{\int }_{{\mathbb{R}}^N}(g(x,u_n)u_n-\theta G(x,u_n))dx}\hfill \\ \hfill & & {\displaystyle +\frac{1}{\parallel u_n{\parallel }^2}{\int }_{{\mathbb{R}}^N}(f(x,u_n)u_n-\theta F(x,u_n))dx}\hfill \\ \hfill & & {\displaystyle \ge (\frac{\theta }{2}-1)+\frac{\mu }{\parallel u_n{\parallel }^2}{\int }_{{\mathbb{R}}^N}(g(x,u_n)u_n-\theta G(x,u_n))dx-(c_1+{\tilde{c}}_1){|w_n|}_2^2}\hfill \\ \hfill & & {\displaystyle \to \frac{\theta }{2}-1,}\hfill \end{array}$$

as $n\to \infty$, which implies that $0\ge {\displaystyle \frac{\theta }{2}}-1$. This is a contradiction with $\theta \ge 4$. Hence, $\left\{u_n\right\}$ is bounded. □

Lemma 6. 

Suppose that $\left(V_1\right)$, $\left(g_1\right)$, and $\left(f_1\right)$ are satisfied. Then, any Palais–Smale sequence for the functional J has a convergent subsequence in E.

Proof. 

By Lemma 5, $\left\{u_n\right\}$ is bounded in E. If necessary going to a subsequence, we can assume that $u_n\rightharpoonup u$ in E. From Lemma 1, we have $u_n\to u$ in $L^p\left(\mathsf{\Omega }\right)$ for all $2\le p<2^{\ast \ast }$. Observe that

$$\begin{array}{ccc}\hfill o\left(1\right)& =& {\displaystyle \langle J^{\prime }\left(u_n\right)-J^{\prime }\left(u\right),u_n-u\rangle }\hfill \\ \hfill & \ge & {\displaystyle \parallel u_n{-u\parallel }^2+b\left({\int }_{{\mathbb{R}}^N}|\nabla u_n{|}^{2}dx-{\int }_{{\mathbb{R}}^N}{|\nabla u|}^{2}dx\right){\int }_{{\mathbb{R}}^N}\nabla u\nabla (u_n-u)dx}\hfill \\ \hfill & & {\displaystyle -{\int }_{{\mathbb{R}}^N}(f(x,u_n)-f(x,u))(u_n-u)dx-\mu {\int }_{{\mathbb{R}}^N}(g(x,u_n)-g(x,u))(u_n-u)dx.}\hfill \end{array}$$

(15)

It is clear that

$${\displaystyle \langle J^{\prime }\left(u_n\right)-J^{\prime }\left(u\right),u_n-u\rangle \to 0,}$$

(16)

as $n\to \infty$. By the boundedness of ${\int }_{{\mathbb{R}}^N}{|\nabla u_n|}^{2}dx$ and $u_n\rightharpoonup u$ in E, we obtain

$${\displaystyle \left({\int }_{{\mathbb{R}}^N}|\nabla u_n{|}^{2}dx-{\int }_{{\mathbb{R}}^N}{|\nabla u|}^{2}dx\right){\int }_{{\mathbb{R}}^N}\nabla u\nabla (u_n-u)dx\to 0,}$$

(17)

as $n\to \infty$. By $\left(f_1\right)$ and Holder’s inequality, we can conclude

$$\begin{array}{ccc}\hfill & & {\displaystyle |{\int }_{{\mathbb{R}}^N}[f(x,u_n)-f(x,u)](u_n-u)\mathrm{d}x|}\hfill \\ \hfill & & {\displaystyle \le {\int }_{{\mathbb{R}}^N}(|f(x,u_n)|+|f(x,u)|)|u_n-u|\mathrm{d}x}\hfill \\ \hfill & & {\displaystyle \le c_0{\int }_{{\mathbb{R}}^N}(|u_n|+|u_n{|}^{p-1}+|u|+{|u|}^{p-1})|u_n-u|\mathrm{d}x}\hfill \\ \hfill & & {\displaystyle \le c_0(|u_n{|}_2|u_n{-u|}_2+|u_n{|}_p^{p-1}|u_n{-u|}_p+{|u|}_2|u_n{-u|}_2+{|u|}_p^{p-1}|u_n-u{|}_p)}\hfill \\ \hfill & & {\displaystyle \le c_0(S_2+S_p^{p-1})(\parallel u_n\parallel |u_n{-u|}_2+\parallel u_n{\parallel }^{p-1}|u_n{-u|}_p+\parallel u\parallel |u_n{-u|}_2+{\parallel u\parallel }^{p-1}|u_n-u{|}_p)}\hfill \\ \hfill & & \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Thus, we obtain that

$$\begin{array}{c}\hfill {\displaystyle {\int }_{{\mathbb{R}}^3}[f(x,u_n)-f(x,u)](u_n-u)\mathrm{d}x\to 0,}\end{array}$$

(18)

as $n\to \infty$. On the other hand, for $i=1,2$, set

$$S_{i,1}=\left({\displaystyle \frac{2^{\ast \ast }}{2^{\ast \ast }-q_i}},{\displaystyle \frac{{22}^{\ast \ast }}{2^{\ast \ast }-2(q_i-1)}}\right],S_{i,2}=\left({\displaystyle \frac{{22}^{\ast \ast }}{2^{\ast \ast }(2-q_i)+2^{\ast \ast }-2}},{\displaystyle \frac{2}{2-q_i}},\right].$$

Then, $S_{i,1}\bigcup S_{i,2}=\left({\displaystyle \frac{2^{\ast \ast }}{2^{\ast \ast }-q_i}},{\displaystyle \frac{2}{2-q_i}},\right]$ and $S_{i,1}\bigcap S_{i,2}\ne \varnothing$. Moreover, let ${\xi }_i=2^{\ast \ast }$ if ${\beta }_i\in S_{i,1}$, ${\xi }_i=2$ if ${\beta }_i\in S_{i,2}\setminus S_{i,1}$. Then, through an easy computation, one obtains that there exists ${\eta }_i\in [2,2^{\ast \ast })$, such that ${\displaystyle \frac{1}{{\beta }_i}}+{\displaystyle \frac{q_i-1}{{\xi }_i}}={\displaystyle \frac{1}{{\eta }_i}}=1$. It follows from (4) and $\left(g_1\right)$ that

$$\begin{array}{ccc}\hfill & & {\displaystyle |{\int }_{{\mathbb{R}}^N}[g(x,u_n)-g(x,u)](u_n-u)\mathrm{d}x|}\hfill \\ \hfill & \le & {\displaystyle {\int }_{{\mathbb{R}}^N}|g(x,u_n)-g(x,u)||u_n-u|\mathrm{d}x}\hfill \\ \hfill & \le & {\displaystyle {\int }_{{\mathbb{R}}^N}(h_1\left(x\right)(|u_n{|}^{q_1-1}+{|u|}^{q_1-1})+h_2\left(x\right)(|u_n{|}^{q_2-1}+{|u|}^{q_2-1}\left)\right)|u_n-u|\mathrm{d}x}\hfill \\ \hfill & \le & {\displaystyle |h_1{|}_{{\beta }_1}(|u_n{|}_{{\xi }_1}^{q_1-1})|u_n{-u|}_{{\eta }_1}+|h_2{|}_{{\beta }_2}(|u_n{|}_{{\xi }_2}^{q_2-1})|u_n{-u|}_{{\eta }_2}}\hfill \\ \hfill & \le & {\displaystyle S_{{\xi }_1}^{q_1-1}|h_1{|}_{{\beta }_1}(\parallel u_n{\parallel }^{q_1-1}+{\parallel u\parallel }^{q_1-1})|u_n{-u|}_{{\eta }_1}+S_{{\xi }_2}^{q_2-1}|h_2{|}_{{\beta }_2}(\parallel u_n{\parallel }^{q_2-1}+{\parallel u\parallel }^{q_2-1})|u_n{-u|}_{{\eta }_2}}\hfill \\ \hfill & \to & 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Thus, we deduce

$$\begin{array}{c}\hfill {\displaystyle {\int }_{{\mathbb{R}}^3}[g(x,u_n)-g(x,u)](u_n-u)\mathrm{d}x\to 0,}\end{array}$$

(19)

as $n\to \infty$. Consequently, (15)–(19) imply that

$$\parallel u_n-u\parallel \to 0,$$

as $n\to \infty$. This completes the proof. □

Proof of Theorem 1. 

Evidently, J is even by $\left(f_4\right)$, $J\left(0\right)=0$. Lemma 3 proves that J satisfies (2) of Theorem 3; Lemma 4 proves that J satisfies (3); Lemma 5 and Lemma 6 prove that J satisfies (4). Hence, by Theorem 3, (1) has infinitely many nontrivial solutions $\left\{u_n\right\}\in E$ and $J\left(u_n\right)\to +\infty$ as $n\to \infty$. This completes the proof. □

4. Proof of Theorem 2

Next, we apply Theorem 4 to obtain infinitely many negative energy solutions. First, we prove that for $\forall c\in \mathbb{R}$, J satisfies ${\left(\mathrm{P}\mathrm{S}\right)}_c^{\ast }$ condition.

Lemma 7. 

Suppose that $\left(V_1\right),\left(f_1\right),\left(f_3^{\prime }\right)$ and $\left(h\right)$ hold. Then, for $\forall c\in \mathbb{R}$, J satisfies the ${\left(PS\right)}_c^{\ast }$ condition.

Proof. 

According to Definition 1, we only need to prove that for any $c\in \mathbb{R}$,

$$u_{n_j}\in Y_{n_j},I\left(u_{n_j}\right)\to c,I{\mid }_{Y_{n_j}}^{\prime }\left(u_{n_j}\right)\to 0,n_j\to \infty$$

has a convergent subsequence. The method of proving is similar to the proof of Lemma 6, we omit it.

Then, in order to obtain infinitely many negative energy solutions, we only need to prove that J satisfies $\left(B_1\right)-\left(B_3\right)$ of Theorem 4.

If $\left(f_1\right)$ holds, then

$$|F(x,u)|\le {\displaystyle \frac{C_0}{2}}{|u|}^2+{\displaystyle \frac{C_0}{p}}{|u|}^p,\forall u\in \mathbb{R},x\in {\mathbb{R}}^N.$$

By Lemma 2, ${\beta }_p\left(k\right)\to 0$, as $k\to \infty$, then for $k_1>0$, when $k>k_1$, one has ${\beta }_p^2\le {\displaystyle \frac{1}{2C_0}}$. For $2<p<2^{\ast \ast }$, there exists $R\in (0,1)$, such that for $R>0$ being small enough, we have

$${\displaystyle \frac{1}{8}}{\parallel u\parallel }^2\ge {\displaystyle \frac{C_0}{p}}{S}_p^p{\parallel u\parallel }^p,\forall u\in E,\parallel u\parallel \le R.$$

Then, by Lemma 2, for $u\in Z_k$, $\parallel u\parallel \le R$, one has

$$\begin{array}{cc}\hfill J\left(u\right)& ={\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{b}{4}}({\int }_{{\mathbb{R}}^N}{|\nabla u|}^2{dx)}^2-{\int }_{{\mathbb{R}}^N}F(x,u)dx-{\displaystyle \frac{\mu }{r}}{\int }_{{\mathbb{R}}^N}h\left(x\right){|u|}^{r}dx\hfill \\ & \ge {\displaystyle \frac{1}{2}}{\parallel u\parallel }^2-{\int }_{{\mathbb{R}}^N}F(x,u)-{\displaystyle \frac{\mu }{r}}{\int }_{{\mathbb{R}}^N}h\left(x\right){|u|}^{r}dx\hfill \\ & \ge {\displaystyle \frac{1}{2}}{\parallel u\parallel }^2-{\displaystyle \frac{C_0}{2}}{\int }_{{\mathbb{R}}^N}{|u|}^{2}dx-{\displaystyle \frac{C_0}{p}}{\int }_{{\mathbb{R}}^N}{|u|}^{p}dx-{\displaystyle \frac{\mu }{r}}{\int }_{{\mathbb{R}}^N}h\left(x\right){|u|}^{r}dx\hfill \\ & \ge {\displaystyle \frac{1}{2}}{\parallel u\parallel }^2-{\displaystyle \frac{C_0}{2}}{\beta }_p^2{\left(k\right)\parallel u\parallel }^2-{\displaystyle \frac{C_0}{p}}{S}_p^p{\parallel u\parallel }^p-{\displaystyle \frac{\mu }{r}}{\beta }_p^r{\left(k\right)|h|}_{{\displaystyle \frac{2}{2-r}}}{\parallel u\parallel }^r\hfill \\ & \ge {\displaystyle \frac{1}{4}}{\parallel u\parallel }^2-{\displaystyle \frac{C_0}{p}}{S}_p^p{\parallel u\parallel }^p-{\displaystyle \frac{\mu }{r}}{\beta }_p^r{\left(k\right)|h|}_{{\displaystyle \frac{2}{2-r}}}{\parallel u\parallel }^r\hfill \\ & \ge {\displaystyle \frac{1}{8}}{\parallel u\parallel }^2-{\displaystyle \frac{\mu }{r}}{\beta }_p^r{\left(k\right)|h|}_{{\displaystyle \frac{2}{2-r}}}{\parallel u\parallel }^r.\hfill \end{array}$$

For $\forall k>k_1$, let $r_k={(4\mu |h|}_{{\displaystyle \frac{2}{2-r}}}{\beta }_p^r{\left(k\right))}^{{\displaystyle \frac{1}{2-r}}}$. Then $r_k\to 0$, as $k\to \infty$. Therefore, there exists $k_0>k_1$, for $k\ge k_0$, $r_k\le R$. Then, for $k\ge k_0$, $u\in Z_k$, $\parallel u\parallel =r_k$, we have $J\left(u\right)\ge 0$. Thus, $\left(B_1\right)$ is proved.

For $u\in Y_k$, $\parallel u\parallel ={\gamma }_k$, where $0<{\gamma }_k<min\{1,r_k\}$. By $\left(f_1\right)$ and $\left(f_3\right)$, there exists $C_1>0$, such that

$$F(x,u)\ge C_1{(|u|}^{\theta }-{|u|}^2),\forall x\in {\mathbb{R}}^N,u\in \mathbb{R}.$$

Then

$$\begin{array}{cc}\hfill J\left(u\right)& ={\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{b}{4}}({\int }_{{\mathbb{R}}^N}{|\nabla u|}^2{dx)}^2-{\int }_{{\mathbb{R}}^N}F(x,u)dx-{\displaystyle \frac{\mu }{r}}{\int }_{{\mathbb{R}}^N}h\left(x\right){|u|}^{r}dx\hfill \\ & \le {\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{b}{4a^2}}{\parallel u\parallel }^4-{\displaystyle \frac{C_1}{\theta }}{\int }_{{\mathbb{R}}^N}{|u|}^{\theta }dx+{\displaystyle \frac{C_1}{2}}{\int }_{{\mathbb{R}}^N}{|u|}^{2}dx-{\displaystyle \frac{\mu }{r}}{\int }_{{\mathbb{R}}^N}{|u|}^{r}dx\hfill \\ & ={\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{b}{4a^2}}{\parallel u\parallel }^4-{\displaystyle \frac{C_1}{\theta }}{|u|}_{\theta }^{\theta }+{\displaystyle \frac{C_1}{2}}{|u|}_2^2-{\displaystyle \frac{\mu }{r}}{|h|}_{{\displaystyle \frac{2}{2-r}}}{|u|}_r^r.\hfill \end{array}$$

As $Y_k$ is a finite dimensional space, and $\theta >4,1<r<2$, then for ${\gamma }_k>0$ being small enough, one has $J\left(u\right)<0$. Then, $\left(B_2\right)$ holds.

By the proving of $\left(B_1\right)$, for $k\ge k_0$, $u\in Z_k$, $\parallel u\parallel \le r_k$, one has

$$J\left(u\right)\ge -{\displaystyle \frac{\mu }{r}}{\beta }_p^r{\left(k\right)|h|}_{{\displaystyle \frac{2}{2-r}}}{\parallel u\parallel }^r\ge -{\displaystyle \frac{\mu }{r}}{\beta }_p^r\left(k\right){|h|}_{{\displaystyle \frac{2}{2-r}}}{r}_k^r.$$

For ${\beta }_p^r\left(k\right)\to 0,r_k\to 0$, as $k\to \infty$. Then, $\left(B_3\right)$ holds.

According to Theorem 4, for any $\mu >0$, (1) has a sequence of negative critical values converging to 0. We finish the proof of Theorem 2. □