Infinitely Many Solutions for a Fractional Schrödinger Equation in $\mathbb {R}^N$ with Combined Nonlinearities

Abstract

This paper is devoted to the following class of nonlinear fractional Schrödinger equations:

$$\begin{aligned} (-\Delta )^{s}u+V(x)u=f(x,u)+\lambda g(x,u), \quad \text {in}\, \mathbb {R}^N, \end{aligned}$$

where $s\in (0,1), \ N>2s, (-\Delta )^{s}$ stands for the fractional Laplacian, $\lambda \in \mathbb {R}$ is a parameter, $V\in C(\mathbb {R}^N,\mathbb {R}), f(x,u)$ is superlinear and g(x, u) is sublinear with respect to u, respectively. We prove the existence of infinitely many high energy solutions of the aforementioned equation by means of the Fountain theorem. Some recent results are extended and sharply improved.

1 Introduction

Consider the following fractional Schrödinger equation

$$\begin{aligned} (-\Delta )^{s}u+ V(x)u=f(x,u), \quad x \in \mathbb {R}^N, \end{aligned}$$

(1.1)

where $s \in (0,1), N>2s$ and $(-\Delta )^{s}$ stands for the fractional Laplacian which can be defined for a sufficiently smooth function u as

$$\begin{aligned} (-\Delta )^{s}u (x)=C(N,s)\lim \limits _{\varepsilon \rightarrow 0^+}\int _{\mathbb {R}^N\setminus B(x,\varepsilon )}\frac{u (x)-u (y)}{| x-y| ^{N+2s}}\textrm{d}y,\quad x\in \mathbb {R}^N, \end{aligned}$$

(1.2)

where $B(x,\varepsilon )=\{x\in \mathbb {R}^N\, :\, |x|< \varepsilon \}$ and $C(N,s)>0$ is a dimensional constant that depends on N and s (see [6]).

Equation (1.1) arises in the study of the following fractional Schrödinger equation

$$\begin{aligned} i\partial _t \Psi (x,t)+(-\Delta )^{s}\Psi (x,t)+(V(x)-\omega )\Psi (x,t)=h(|\Psi (x,t)|)\Psi (x,t), \end{aligned}$$

when looking for standing waves, that is, solutions of the form $\Psi (x,t)=\exp (-i\omega t)u(x)$. The fractional Schrödinger equation was introduced by Laskin [14, 15] in the context of fractional quantum mechanics, as a result of extending the Feynman path integral from the Brownian-like to Lévy-like quantum mechanical paths. It is also appeared in several subjects such as plasma physics, image processing, finance and stochastic models, see for instance [1, 4, 10, 16].

In recent years, Eq. (1.1) has been extensively studied under various assumptions on V and f and there are many interesting results in the literature on the existence and multiplicity of solutions to problem (1.1) has been obtained via variational approaches, we refer the readers to [3, 5, 7,8,9, 11,12,13, 17, 19,20,23]. In particular, the existence of infinitely many high or small energy solutions to problem (1.1) was established in [5, 7, 9, 11,12,13, 17, 20] by the aid of variant fountain theorems (see [25]) or the symmetric mountain pass theorem (see [24]). However, there are few papers concern with the existence of infinitely many (high or small) energy solutions to problem (1.1) in the case where f(x, u) is a combination of sublinear and superlinear terms at infinity with respect to u, see for instance [7, 17, 21].

In [7], Du and Tian considered the following class of fractional Schrödinger equations with concave and critical nonlinearities

$$\begin{aligned} (-\Delta )^{s} u+V(x) u=\mu a(x)|u|^{q-2} u+|u|^{2_{s}^{*}-2} u, \quad x \in \mathbb {R}^{N}, \end{aligned}$$

(1.3)

where (and in the sequel) $2_s^*=\frac{2N}{N-2s}$ is the critical Sobolev exponent, $\mu >0$ is a parameter, $1<q<2, a(x)$ is positive continuous functions satisfying $a(x) \in L^{\frac{2}{2-q}}\left( \mathbb {R}^{N}\right) \cap L^{\frac{2_{s}^{2}}{2 s}-q}\left( \mathbb {R}^{N}\right)$ and V(x) satisfies the following assumptions

(V):

$V \in C(\mathbb {R}^N,\mathbb {R})$ satisfies $\inf _{x\in \mathbb {R}^{N}} V(x)\ge V_{0}>0$, where $V_{0}$ is a constant. Moreover, there exists $r_0>0$ such that

$$\begin{aligned} \lim _{|y|\rightarrow \infty } {{\,\textrm{meas}\,}}\{x \in \mathbb {R}^{N} \, : \,|x-y|\le r_0,\, V(x)\le M\}=0,\quad \forall M>0, \end{aligned}$$

where ${{\,\textrm{meas}\,}}(.)$ is the Lebesgue measure on $\mathbb {R}^N$. The authors proved that there exists $\mu ^*> 0$ such that, for any $0< \mu < \mu ^*$, problem (1.3) possesses infinitely many small energy solutions by using the Dual fountain theorem.

In [21], Timoumi established infinitely many small energy solutions to the problem

$$\begin{aligned} (-\Delta )^{s}u+ V(x)u=g(x,u)+h(x,u), \quad \text {in}\, \mathbb {R}^N, \end{aligned}$$

by means of the Dual Fountain Theorem (see [24]), where V(x) satisfies assumptions (V), g(x, u) is sublinear in u and h(x, u) is superlinear in u.

Li and Shang [17] studied the following problem

$$\begin{aligned} (-\Delta )^{s} u+V(x) u=f(x, u)+\lambda h(x)|u|^{p-2} u,\quad x \in \mathbb {R}^{N} \end{aligned}$$

(1.4)

where $\lambda >0$ is a parameter, $p\in [1,2), h \in L^{\frac{2}{2-p}}\left( \mathbb {R}^{N}\right)$ and V and f satisfies the following assumptions:

$(V')$:

$V(x)\in C\left( \mathbb {R}^{N},\mathbb {R}\right)$, $\inf _{x\in \mathbb {R}^{N}}V(x)\ge V_{0}>0$ and $\lim _{|x| \rightarrow \infty } V(x)=\infty$;

$(f_{1})$:

$f \in C(\mathbb {R}^{N}\times \mathbb {R},\mathbb {R})$ and there exist constants $a_{1}, a_{2} \ge 0, q \in \left[ 2, \frac{2 N+4 s}{N}\right)$ with $\frac{a_{1}}{2 S_{2}^{2}}+\frac{a_{2}}{q S_{q}^{q}}<\frac{1}{2}$ such that

$$\begin{aligned} |f(x, u)| \le a_{1}|u|+a_{2}|u|^{q-1}, \quad \forall (x, u) \in \mathbb {R}^{N} \times \mathbb {R}, \end{aligned}$$

where $S_{q}$ is the best constant for the embedding of $X \subset L^{q}\left( \mathbb {R}^{N}\right)$ and

$$\begin{aligned} X=\left\{ u \in L^{2}\left( \mathbb {R}^{N}\right) : \int _{\mathbb {R}^{N}} \int _{\mathbb {R}^{N}} \frac{|u(x)-u(z)|^{2}}{|x-z|^{N+2 s}} \textrm{d} x \textrm{d} z+\int _{\mathbb {R}^{N}} V(x) u(x)^{2} \textrm{d} x<+\infty \right\} ; \end{aligned}$$

$(f_{2})$:

$\lim \limits _{t\rightarrow \infty }\frac{F(x,t)}{\vert t\vert ^{2}}=\infty$ uniformly in $x\in \mathbb {R}^{N}$ and there exists $r_{1}>0$ such that $F(x, u) \ge 0$, for any $x \in \mathbb {R}^{N}, u \in \mathbb {R}$ and $|u| \ge r_{1}$, where $F(x,t)=\int _{0}^{t}f(x,s)\textrm{d}s$;

$(f_{3})$:

$2F(x,u) < f(x,u)u, \forall (x,u)\in \mathbb {R}^{N}\times \mathbb {R}$.

$(F_{4})$:

$f(x,-u)=-f(x,u)$ for all $(x,u) \in \mathbb {R}^N\times \mathbb {R}$

By using the symmetric mountain pass theorem, the authors showed that there exists a constant $\lambda _0> 0$ such that, for any $\lambda \in (0,\lambda _0)$, problem (1.4) possesses infinitely many high energy solutions.

Motivated by these works, in the present paper we are concerned with the existence of infinitely many high energy solutions to the following class of fractional Schrödinger equation

$$\begin{aligned} (-\Delta )^{s}u+ V(x)u=f(x,u)+\lambda g(x,u), \quad \text {in}\, \mathbb {R}^N, \end{aligned}$$

(1.5)

where $\lambda \in \mathbb {R}$ is a parameter, V(x) satisfies assumptions (V) and f and g satisfy the following assumptions

$(F_1)$:

$f\in C(\mathbb {R}^N\times \mathbb {R},\mathbb {R})$ and there exist constants $c_1,c_2>0$ and $p\in (2,2_s^*)$ such that

$$\begin{aligned} |f(x,u)|\le c_1|u|+c_2|u|^{p-1},\quad \forall (x,u) \in \mathbb {R}^N\times \mathbb {R}, \end{aligned}$$

where $2_s^*=\frac{2N}{N-2s}$ is the critical Sobolev exponent.

$(F_2)$:

$\lim \limits _{|u|\rightarrow \infty } \frac{F(x,u)}{u^2}=+\infty$ a.e. $x\in \mathbb {R^{N}}$, where $F(x,u)=\int _0^u f(x,t)\textrm{d}t$ and there exists $r_1>0$ such that

$$\begin{aligned} \inf _{x\in \mathbb {R}^{N},|u| \ge r_1} F(x,u)\ge 0; \end{aligned}$$

$(F_3)$:

There exist constants $\mu>2, c_3>0$ and $a_0>0$ such that such that

$$\begin{aligned} \mu F(x,u) \le f(x,u)u+c_3 |u|^2,\quad \forall (x,|u|) \in \mathbb {R}^N \times [a_0,\infty ). \end{aligned}$$

$(g_1)$:

There exist constants $1<\delta _1<\delta _2<2$ and positive functions $\xi _i \in L^{\frac{2}{2-\delta _i}}(\mathbb {R}^N)$ $(i=1,2)$ such that

$$\begin{aligned} |g(x,u)|\le \xi _1(x) |u|^{\delta _1-1} +\xi _2 (x)|u|^{\delta _2-1},\quad \forall (x,u) \in \mathbb {R}^N\times \mathbb {R}. \end{aligned}$$

$(g_2)$:

$g(x,-u)=-g(x,u)$ for all $(x,u) \in \mathbb {R}^N\times \mathbb {R}$;

By using the Fountain theorem (i.e., [24, Theorem 3.6]), we prove the existence of an unbounded sequence of nontrivial solutions $\{u_k\}$ to problem (1.5) under assumptions (V), $(F_1)$–$(F_4)$ and $(g_1)$–$(g_2)$. Our result extends and sharply improves that in [17].

The remainder of this paper is organized as follows. In Sect. 2, we prepare the variational framework of the studied problem. In Sect. 3, employing the fountain theorem (3.1), we establish the existence of infinitely many high energy solutions to problem (1.5).

2 Variational Setting and Main Result

In this section, for the reader’s convenience, we shall introduce some notations and we revise some known results about the fractional Sobolev spaces which can be found in [6].

As usual, for $1\le p<+\infty$, we define

$$\begin{aligned} \Vert u\Vert _{L^p}:=\Vert u\Vert _{p}=\left( \int _{\mathbb {R}^N}| u|^{p}\textrm{d}x\right) ^{\frac{1}{p}},\quad u\in L^p(\mathbb {R}^N), \end{aligned}$$

The fractional Sobolev space $H^{s} (\mathbb {R}^N)=W^{s,2}(\mathbb {R}^N)$ is defined by

$$\begin{aligned} H^{s} (\mathbb {R}^N):=\left\{ {u\in L^{2}(\mathbb {R}^N)\, :\, \frac{| u(x)-u(y)|}{| x-y|^{\frac{N}{2}+s}}\in L^{2}\left( \mathbb {R}^N\times \mathbb {R}^N\right) }\right\} \end{aligned}$$

with the inner product and the norm

$$\begin{aligned} \langle u,v\rangle _{H^{s }}= & {} \iint _{\mathbb {R}^N\times \mathbb {R}^N}\frac{(u(x)-u(y))(v(x)-v(y))}{| x-y|^{N+2s}}\textrm{d}x\textrm{d}y+\int _{\mathbb {R}^N}u(x)v(x)\textrm{d}x,\\ \Vert u\Vert _{H^{s}}^{2}= & {} \langle u,u\rangle _{H^{s}}=\iint _{\mathbb {R}^N\times \mathbb {R}^N}\frac{| u(x)-u(y)|^{2}}{| x-y|^{N+2s}}\textrm{d}x\textrm{d}y+\int _{\mathbb {R}^N}| u(x)|^{2} \textrm{d}x, \end{aligned}$$

where the norm

$$\begin{aligned}{}[u]_{H^{s}}^{2}=\iint _{\mathbb {R}^N\times \mathbb {R}^N}\frac{| u(x)-u(y)|^{2}}{| x-y|^{N+2s}}\textrm{d}x\textrm{d}y \end{aligned}$$

is the so-called Gagliardo semi-norm of u.

Let ${\mathscr {S}}(\mathbb {R}^N)$ be the Schwartz space of rapidly decaying $C^{\infty }$ functions in $\mathbb {R}^N$. We recall that the Fourier transform of a function $u \in {\mathscr {S}} (\mathbb {R}^N)$ is defined as

$$\begin{aligned} {\mathscr {F}}u(\xi ):= \frac{1}{(2\pi )^{N}}\int _{\mathbb {R}^N}e^{- i x\xi }u (x)\textrm{d}x. \end{aligned}$$

By Plancherel’s theorem, we have

$$\begin{aligned} \Vert u \Vert _{2}=\Vert {\mathscr {F}} u \Vert _{2}, \quad \forall u \in {\mathscr {S}}(\mathbb {R}^N). \end{aligned}$$

Let $s \in (0,1)$, the fractional Laplacian $(-\Delta )^{s}$ of a function $u \in {\mathscr {S}}(\mathbb {R}^N)$ is defined by means of the Fourier transform as

$$\begin{aligned} {\mathscr {F}}\left( (-\Delta )^{s}u\right) (\xi )=|\xi |^{2s}{\mathscr {F}}u (\xi ),\quad \forall s \in (0,1). \end{aligned}$$

The space $H^{s}(\mathbb {R}^N)$ can also be described via the Fourier transform as follows

$$\begin{aligned} H^{s} (\mathbb {R}^N):=\left\{ {u\in L^{2}(\mathbb {R}^N)\, :\, \int _{\mathbb {R}^N}\left( 1+| \xi |^{2}\right) ^{s} |{\mathscr {F}}u (\xi )|^{2} \textrm{d}\xi <\infty }\right\} , \end{aligned}$$

and the norm is defined by

$$\begin{aligned} \Vert u\Vert _{H^{s }}=\left( \int _{\mathbb {R}^N}\left( 1+| \xi |^{2}\right) ^{s} | {\mathscr {F}} u (\xi )|^{2} \textrm{d}\xi \right) ^{\frac{1}{2}}. \end{aligned}$$

For the problem (1.5), we define the following Hilbert space

$$\begin{aligned} H:=\left\{ u \in H^s( \mathbb {R}^N)\, : \, \int _{\mathbb {R}^N} V(x) |u|^{2} \textrm{d}x <\infty \right\} , \end{aligned}$$

endowed with the inner product

$$\begin{aligned} \langle u,v\rangle {:=} \langle u,v\rangle _{H}= & {} \iint _{\mathbb {R}^N\times \mathbb {R}^N}\frac{(u(x)-u(y))(v(x)-v(y))}{| x-y|^{N+2s}}\textrm{d}x\textrm{d}y\\{} & {} +\int _{\mathbb {R}^N}V(x)u(x)v(x)\textrm{d}x. \end{aligned}$$

Then, the norm on H is given by

$$\begin{aligned} \Vert u\Vert :=\left( \iint _{\mathbb {R}^N\times \mathbb {R}^N}\frac{| u(x)-u(y)|^{2}}{| x-y|^{N+2s}}\textrm{d}x\textrm{d}y+\int _{\mathbb {R}^N}V(x)|u|^{2} \textrm{d}x\right) ^{\frac{1}{2}}. \end{aligned}$$

Obviously, by assumptions (V), the embedding $H \hookrightarrow H^s( \mathbb {R}^N)$ is continuous.

From [6], the embeddings $H^s(\mathbb {R}^N) \hookrightarrow L^p(\mathbb {R}^N)$ is continuous for $p \in [2,2_{s}^{*}]$. Therefore, $H \hookrightarrow L^p(\mathbb {R}^N),\, 2\le p\le 2_{s}^{*}$ is continuous, namely, there exist constants $\eta _p>0$ such that

$$\begin{aligned} \Vert u\Vert _{p}\le \eta _p \Vert u\Vert ,\quad \forall u \in H,\, p\in [2,2_s^*], \end{aligned}$$

(2.1)

Moreover, from [20], we know that the embedding $H \hookrightarrow L^p(\mathbb {R}^N)$ is compact for $2 \le p< 2_s^{*}$ under condition (V).

For the fractional Schrödinger equation (1.5), the associated energy functional is defined on H as follows

$$\begin{aligned} \begin{aligned} I_\lambda (u)=\frac{1}{2}\Vert u\Vert ^2-\int _{\mathbb {R}^N}F(x,u)\textrm{d}x-\lambda \int _{\mathbb {R}^N}G(x,u)\textrm{d}x. \end{aligned} \end{aligned}$$

(2.2)

By hypotheses $(V), (F_1)$ and $(g_1)$, the functional I is well define and of class $C^1(H,\mathbb {R})$ with

$$\begin{aligned} \langle I_\lambda ^{\prime }(u),v\rangle= & {} \int _{\mathbb {R}^{2N}}\frac{(u(x)-u(y))(v(x)-v(y))}{| x-y|^{N+2s}}\textrm{d}x\textrm{d}y\nonumber \\{} & {} +\int _{\mathbb {R}^N}V(x) u v \textrm{d}x-\int _{\mathbb {R}^N}f(x,u)v \textrm{d}x-\lambda \int _{\mathbb {R}^N}g(x,u)v \textrm{d}x, \end{aligned}$$

(2.3)

for all $v\in H$. Besides, the critical points of I in H are solutions of problem (1.5). Now, we are ready to state the main result of this paper as follows.

Theorem 2.1

Assume that conditions $(V), (F_1)$–$(F_4)$ and $(g_1)$–$(g_2)$ hold. Then there exists ${\overline{\lambda }}>0$ such that problem (1.5) possesses infinitely many nontrivial solutions $\{u_k\}$ provided $|\lambda |\le {\overline{\lambda }}$. Moreover, there holds

$$\begin{aligned} I_\lambda (u_k) \rightarrow \infty \text { as } k \rightarrow \infty . \end{aligned}$$

Remark 2.1

Since the problem (1.5) is defined on the entire space $\mathbb {R}^N$, the main difficulty of this problem is the lack of compactness for Sobolev embedding theorem. In the context of studying of the existence of solutions for the classical Schrödinger equation

$$\begin{aligned} -\Delta u+V(x) u=f(x, u),\quad x \in R^{N}, \end{aligned}$$

Bartsch et al. [2] presented the general conditions (V) which guarantee the compactness of the embeddings $\left\{ u \in H^{1}\left( R^{N}\right) \,:\, \int _{R^{N}} V(x) |u|^{2} d x<+\infty \right\} \hookrightarrow L^p\left( \mathbb {R}^{N}\right) ,\, p\in [2,\frac{2N}{N-2}]$. Furthermore, conditions (V) are weaker than the coercivity condition $(V')$ used in [17].

Remark 2.2

Firstly, comparing with Theorem 1.1 in [17], our assumptions $(F_1)$–$(F_3)$ are more general than $(f_1)$–$(f_3)$. Indeed, let $f(x,u)=au+b|u|^{p-2}u$, where $a>2S_2^2, b > qS_q^q$ and $p\in (2,2_2^*)$. Then, clearly f satisfies $(F_1)$ but not $(f_1)$ since $\frac{a}{2 S_{2}^{2}}+\frac{b}{q S_{q}^{q}}>2$. Secondly, let

$$\begin{aligned} f(x,t)= 3 |t|t-\frac{15}{2} |t|^{1 / 2}t+t,\quad t\in \mathbb {R}. \end{aligned}$$

Then,

$$\begin{aligned} F(x,t)= |t|^{3}-3 |t|^{5 / 2}+\frac{1}{2}t^2. \end{aligned}$$

It is easy to verify that the above function f satisfies $(F_1)$, $(F_2)$, $(F_4)$ and $(F_3)$ (which was initially gave in [18]) with $\mu =\frac{5}{2}$. However, f does not satisfy $(f_3)$, in fact we have

$$\begin{aligned} f(x,t)t-2 F(x,t) = |t|^{3}-\frac{3}{2}|t|^{5 / 2} \le 0,\quad \forall t \in \left[ -\frac{9}{4},\frac{9}{4}\right] . \end{aligned}$$

This shows that $(f_3)$ is not satisfied. Finally, it is easy to see that ${\widetilde{g}}(x, u)=\lambda h(x)|u|^{p-2} u$ considered in (1.4) is a special case of g(x, u) considered in this paper. Furthermore, unlike (1.4), the parameter $\lambda$ in (1.5) is allowed to be sign-changing. Consequently, Theorem 2.1 generalizes and sharply improves Theorem 1.1 in [17].

Remark 2.3

When $s = 1$, Eq. (1.5) becomes the classical Schrödinger equation

$$\begin{aligned} -\Delta u+V(x) u=f(x, u)+\lambda g(x,u),\quad x \in R^{N}, \end{aligned}$$

As far as we know, our result is new even for the case $s=1$.

3 Proof of the Main Result

Hereafter, we shall use $c_i,C_i,\, i=1,2,\ldots$ to denote various positive constants which may change from line to line. We start this section by introducing some variational preliminaries and abstract results that we need to prove our main results.

Definition 3.1

((PS)-condition)

• A sequence $\{u_{n}\}\subset H$ is said to be a Palais–Smale sequence at level $c\in \mathbb {R}$ ((PS)$_{c}$ sequence for short) if $I(u_{n})\rightarrow c$ and $I^{\prime }(u_{n})\rightarrow 0$ in $H^*$ the dual space of H.

• The functional I satisfies the Palais–Smale condition at the level c ((PS)$_{c}$ condition for short) if any (PS)$_{c}$ sequence has a convergent subsequence.

Lemma 3.1

Under the assumptions of Theorem 2.1, the functional $I_\lambda$ satisfies the (PS)$_{c}$ condition for any $c>0$.

Proof

Let $\{u_n\}\subset H$ be any (PS) sequence of $I_\lambda$, that is,

$$\begin{aligned} I_\lambda (u_n) \rightarrow c >0 ,\quad I_\lambda '(u_n)\rightarrow 0 \text { in } H^*. \end{aligned}$$

(3.1)

First, we prove that $\{u_n\}$ is bounded in H. Arguing indirectly, suppose that $\Vert u_n\Vert \rightarrow \infty$ as $n\rightarrow \infty$. Set $v_n = \frac{u_n}{\Vert u_n\Vert }$, then $\Vert v_n\Vert =1$, thus $\{v_n\}$ is bounded in H. Using assumption $(F_1)$ we have

$$\begin{aligned} \begin{aligned} |F(x,u)| =\left| F(x,u)-F(x,0)\right|&=\left| \int _{0}^{1}f(x,tu)u\textrm{d}t\right| \\&\le \int _{0}^{1}\left( c_1|u|^2 t+c_2|u|^p t^{p-1}\right) \textrm{d}t\\&=\frac{c_1}{2}|u|^2+\frac{c_2}{p}|u|^p,\qquad \forall (x,u)\in \mathbb {R}^N\times \mathbb {R}. \end{aligned} \end{aligned}$$

(3.2)

Set ${\mathcal {F}}(x,u_n)=f(x,u_n)u_n-\mu F(x,u_n)$. Therefore, for $x\in \mathbb {R}^{N}$ and $|u(x)| < a_0$, by (3.2), we have

$$\begin{aligned} \begin{aligned} \left| f(x,u) u- \mu F(x,u)\right|&\le \vert f(x,u) u\vert +\mu \vert F(x,u)\vert \\&\le \left( c_{1}\vert u\vert ^{2}+c_{2}\vert u \vert ^{p}\right) +\left( c_{1} \frac{\mu }{2}\vert u\vert ^{2}+c_{2}\frac{\mu }{p}\vert u\vert ^{p}\right) \\&\le \left( \frac{2+\mu }{2} c_{1}+\frac{p+\mu }{p}c_{2}a_0^{p-2}\right) \vert u\vert ^{2}\\&= c_{3}\vert u \vert ^{2}, \end{aligned} \end{aligned}$$

where $\mu$ and $a_0>0$ are given in $(F_3)$. Combining the above inequality with $(F_3)$, we conclude that there exists $c_4>0$ such that

$$\begin{aligned} {\mathcal {F}}(x,u)= f(x,u) u- \mu F(x,u)\ge -c_4|u|^2,\quad \forall (x,u)\in \mathbb {R}^{N}\times \mathbb {R}. \end{aligned}$$

(3.3)

By (2.2), (2.3) and (3.3) , we have

$$\begin{aligned} \begin{aligned} \mu I_\lambda (u_n)-\langle I_\lambda ^{\prime }(u_n),u_n\rangle =&\frac{\mu -2}{2}\Vert u_n\Vert ^2+\int _{\mathbb {R}^N}{\mathcal {F}}(x,u_n)\textrm{d}x-\lambda \int _{\mathbb {R}^N}{\mathcal {G}}(x,u_n)\textrm{d}x \\ \ge&\frac{\mu -2}{2}\Vert u_n\Vert ^2 -c_4 \int _{\mathbb {R}^N} |u_n|^2 \textrm{d}x-\lambda \int _{\mathbb {R}^N}{\mathcal {G}}(x,u_n)\textrm{d}x, \end{aligned} \nonumber \\ \end{aligned}$$

(3.4)

where ${\mathcal {G}}(x,u):=g(x,u)u-\mu G(x,u)$. By $(g_1)$ one has

$$\begin{aligned} \begin{aligned} |{\mathcal {G}}(x,u)|&=|g(x,u)u-\mu G(x,u)| \\&\le |g(x,u)u|+\mu |G(x,u)|\\&\le \xi _1(x)|u|^{\delta _1}+\xi _2(x)|u|^{\delta _2}+\frac{\mu }{\delta _1}\xi _1(x)|u|^{\delta _1}+\frac{\mu }{\delta _2}\xi _2(x)|u|^{\delta _2}\\&:=\gamma _1 \xi _1(x)|u|^{\delta _1}+\gamma _2\xi _2(x)|u|^{\delta _2}, \end{aligned} \end{aligned}$$

(3.5)

where $\gamma _i=\frac{\mu + \delta _i}{\delta _i}$ ($i=1,2$). Since $\xi _i \in L^{\frac{2}{2-\delta _i}}(\mathbb {R}^N)$, it follows from (3.5), the Hölder’s inequality and (2.1)

$$\begin{aligned} \begin{aligned}&\left| \int _{\mathbb {R}^N}{\mathcal {G}}(x,u_n)\textrm{d}x\right| \\&\le \int _{\mathbb {R}^N} |{\mathcal {G}}(x,u_n)|\textrm{d}x\\&\le \gamma _1 \int _{\mathbb {R}^N} \xi _1(x)|u_n|^{\delta _1}\textrm{d}x +\gamma _2 \int _{\mathbb {R}^N} \xi _2(x)|u_n|^{\delta _2}\textrm{d}x\\&\le \sum _{i=1}^2 \gamma _i \left( \int _{\mathbb {R}^N}|\xi _i(x)|^{\frac{2}{2-\delta _i}}\textrm{d}x\right) ^{\frac{2-\delta _i}{2}}\left( \int _{\mathbb {R}^N}|u_n|^2\textrm{d}x\right) ^{\frac{\delta _i}{2}}\\&\le \gamma _1 \Vert \xi _1\Vert _{\frac{2}{2-\delta _1}}\Vert u_n\Vert _{2}^{\delta _1}+\gamma _2 \Vert \xi _2\Vert _{\frac{2}{2-\delta _2}}\Vert u_n\Vert _{2}^{\delta _2}\\&\le \gamma _1 \eta _2^{\delta _1} \Vert \xi _1\Vert _{\theta _1}\Vert u_n\Vert ^{\delta _1}+\gamma _2\eta _2^{\delta _2} \Vert \xi _2\Vert _{\theta _2}\Vert u_n\Vert ^{\delta _2}\\&:= C_1 \Vert \xi _1\Vert _{\theta _1}\Vert u_n\Vert ^{\delta _1}+C_2 \Vert \xi _2\Vert _{\theta _2}\Vert u_n\Vert ^{\delta _2}, \end{aligned} \end{aligned}$$

(3.6)

where $C_i= \gamma _i \eta _2^{\delta _i}$ and $\theta _i =\frac{2}{2-\delta _i}, i=1,2$. Combining (3.1) with (3.4) and (3.6), for sufficiently large $n\in \mathbb {N}$, there exists a constant $C_3>0$ such that

$$\begin{aligned} \begin{aligned} C_3&\ge \mu I_\lambda (u_n)-\langle I_\lambda ^{\prime }(u_n),u_n\rangle \\&\ge \frac{\mu -2}{2}\Vert u_n\Vert ^2 -c_4 \int _{\mathbb {R}^N} |u_n|^2 \textrm{d}x-\lambda \int _{\mathbb {R}^N}{\mathcal {G}}(x,u_n)\textrm{d}x\\&\ge \frac{\mu -2}{2}\Vert u_n\Vert ^2 -c_4 \Vert u_n\Vert _2^2-|\lambda |\left( C_1 \Vert \xi _1\Vert _{\theta _1}\Vert u_n\Vert ^{\delta _1}+C_2 \Vert \xi _2\Vert _{\theta _2}\Vert u_n\Vert ^{\delta _2}\right) , \end{aligned} \end{aligned}$$

which yields

$$\begin{aligned} \frac{\Vert u_n\Vert _2^2}{\Vert u_n\Vert ^2}\ge & {} \frac{\mu -2}{2c_4}-\frac{1}{c_4}\left[ \frac{C_3}{\Vert u_n\Vert ^2}+|\lambda |\frac{C_1 \Vert \xi _1\Vert _{\theta _1}}{\Vert u_n\Vert ^{2-\delta _1}}\right. \\{} & {} \left. +|\lambda |\frac{C_2 \Vert \xi _2\Vert _{\theta _2}}{\Vert u_n\Vert ^{2-\delta _2}}\right] . \end{aligned}$$

Since $1<\delta _1<\delta _2<2$ and $\Vert u_n\Vert \rightarrow \infty$, we can choose a large $n\in \mathbb {N}$ so that

$$\begin{aligned} \frac{C_3}{\Vert u_n\Vert ^2}+|\lambda |\frac{C_1 \Vert \xi _1\Vert _{\theta _1}}{\Vert u_n\Vert ^{2-\delta _1}}+|\lambda |\frac{C_2 \Vert \xi _2\Vert _{\theta _2}}{\Vert u_n\Vert ^{2-\delta _2}}\le \frac{\mu -2}{4}, \end{aligned}$$

we then conclude

$$\begin{aligned} \Vert v_n\Vert _2^2=\frac{\Vert u_n\Vert _2^2}{\Vert u_n\Vert ^2}\ge \frac{\mu -2}{4 c_4}>0. \end{aligned}$$

(3.7)

Set $\Omega _n=\{x \in \mathbb {R}^{N}\, :\, |u_n(x)|\le r_1 \}$ and $A_n=\{x \in \mathbb {R}^{N}\, :\, v_n(x)\ne 0\}$, then ${{\,\textrm{meas}\,}}(A_n)>0$ due to (3.7). Besides, since $\Vert u_n\Vert \rightarrow \infty$ as $n\rightarrow \infty$, we obtain

$$\begin{aligned} |u_n(x)|\rightarrow \infty \quad \text {as}\quad n\rightarrow \infty ,\quad \forall x\in A_n. \end{aligned}$$

(3.8)

Hence, $A_n\subseteq \mathbb {R}^N \setminus \Omega _n$ for $n\in \mathbb {N}$ large enough.

Similarly to (3.6), by $(g_1)$, (2.1) and Hölder’s inequality, we derive that

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^N} G(x,u_n)\textrm{d}x&\le \int _{\mathbb {R}^N} \xi _1(x)|u_n|^{\delta _1}\textrm{d}x + \int _{\mathbb {R}^N} \xi _2(x)|u_n|^{\delta _2}\textrm{d}x\\&\le \Vert \xi _1\Vert _{\theta _1}\Vert u_n\Vert _2^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\Vert u_n\Vert _2^{\delta _2}\\&\le \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}\Vert u_n\Vert ^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\Vert u_n\Vert ^{\delta _2} \end{aligned} \end{aligned}$$

(3.9)

Therefore

$$\begin{aligned} \int _{\mathbb {R}^{N}}\frac{|G(x,u_n)|}{\Vert u_n\Vert ^2}\textrm{d}x\le \frac{\Vert \xi _1\Vert _{\theta _1}\Vert u_n\Vert ^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\Vert u_n\Vert ^{\delta _2}}{\Vert u_n\Vert ^2} \longrightarrow 0, \quad \text { as } n\rightarrow \infty , \end{aligned}$$

(3.10)

in view of $\Vert u_n\Vert \rightarrow \infty$ and $1< \delta _1<\delta _2<2$. Hence, by (3.2), (2.1), (2.2), (3.1), (3.8), (3.10) and Fatou’s lemma, we obtain

$$\begin{aligned} 0&=\lim \limits _{n\rightarrow \infty }\frac{I_\lambda (u_n)}{\Vert u_n\Vert ^2} \nonumber \\&=\lim \limits _{n\rightarrow \infty }\left[ \frac{1}{2}-\int _{\mathbb {R}^{N}}\frac{F(x,u_n)}{\Vert u_n\Vert ^2}\textrm{d}x-\lambda \int _{\mathbb {R}^{N}}\frac{G(x,u_n)}{\Vert u_n\Vert ^2}\textrm{d}x\right] \nonumber \\&= \frac{1}{2}+\lim \limits _{n\rightarrow \infty }\left[ -\int _{\Omega _n}\frac{F(x,u_n)}{u_n^2}v_n^2 \textrm{d}x-\int _{\mathbb {R}^{N}\setminus \Omega _n}\frac{F(x,u_n)}{u_n^2} v_n^2 \textrm{d}x\right] \nonumber \\&\le \frac{1}{2}+\limsup \limits _{n\rightarrow \infty }\left[ \left( c_1+c_2 r_1^{p-2}\right) \int _{\mathbb {R}^{N}} |v_n|^2\textrm{d}x -\int _{\mathbb {R}^{N}\setminus \Omega _n}\frac{F(x,u_n)}{u_n^2} v_n^2 \textrm{d}x\right] \nonumber \\&\le \frac{1}{2}+\left( c_1+c_2 r_1^{p-2}\right) \eta _2^2-\liminf \limits _{n \rightarrow \infty }\int _{\mathbb {R}^{N}\setminus \Omega _n}\frac{F(x,u_n)}{u_n^2} v_n^2 \textrm{d}x \nonumber \\&\le C_4-\int _{A_n}\liminf \limits _{n \rightarrow \infty }\frac{F(x,u_n)}{u_n^2} v_n^2 \textrm{d}x \nonumber \\&= C_4- \int _{\mathbb {R}^{N}}\liminf \limits _{n \rightarrow \infty }\frac{F(x,u_n)}{u_n^2}[\chi _{A_n}(x)]v_n^2 \textrm{d}x \longrightarrow -\infty , \quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

(3.11)

This is an obvious contradiction. Consequently, $\{u_n\}$ is bounded in H.

Since $\{u_n\}$ is bounded in H, then there exists a constant $M>0$ such that

$$\begin{aligned} \Vert u_n\Vert \le M, \quad \forall n\in \mathbb {N}. \end{aligned}$$

(3.12)

Furthermore, passing to a subsequence, there is $u \in H$ such that

$$\begin{aligned} \begin{aligned}&u_n \rightharpoonup u \text { in } H;\\&u_n \rightarrow u\text { in } L^p(\mathbb {R}^N),\quad 2\le p<2_s^*;\\&u_n \rightarrow u \text { a.e. in } \mathbb {R}^N. \end{aligned} \end{aligned}$$

(3.13)

By $(F_1)$, (2.1), (3.12), the Hölder’s inequality and (3.13), it has

$$\begin{aligned}&\int _{\mathbb {R}^N}\left| f(x,u_n)-f(x,u)\right| (u_n-u) \textrm{d}x \nonumber \\&\quad \le \int _{\mathbb {R}^N} |f(x,u_n)|(u_n-u) \textrm{d}x + \int _{\mathbb {R}^N} |f(x,u)|(u_n-u) \textrm{d}x \nonumber \\&\quad \le c_1\int _{\mathbb {R}^N}\left( |u_n|+|u|\right) (u_n-u)\textrm{d}x+c_2 \int _{\mathbb {R}^N}(|u_n|^{p-1}+|u|^{p-1})(u_n-u)\textrm{d}x \nonumber \\&\quad \le C_5 \Vert u_n-u\Vert _2+C_6\Vert u_n-u\Vert _p=o_n(1), \end{aligned}$$

(3.14)

where $C_5=c_1(\eta _2M+\Vert u\Vert _2), C_6=c_2 \left( \eta _p^{p-1}M^{p-1}+\Vert u\Vert _p^{p-1}\right)$ and $o_n(1)\rightarrow 0$ as $n\rightarrow \infty$.

On the other hand, it follows from $(g_1)$, (2.1), (3.12), Hölder’s inequality and (3.13) that

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^N}|g(x,u_n)-g(x,u)|(u_n-u)\textrm{d}x \\&\quad \le \int _{\mathbb {R}^N} |g(x,u_n)|(u_n-u)\textrm{d}x+\int _{\mathbb {R}^N} |g(x,u)|(u_n-u)\textrm{d}x\\&\quad \le \int _{\mathbb {R}^N} \xi _1(x)|u_n|^{\delta _1-1}(u_n-u)\textrm{d}x + \int _{\mathbb {R}^N} \xi _2(x)|u_n|^{\delta _2-1}(u_n-u)\textrm{d}x\\&\qquad + \int _{\mathbb {R}^N} \xi _1(x)|u|^{\delta _1-1}(u_n-u)\textrm{d}x + \int _{\mathbb {R}^N} \xi _2(x)|u|^{\delta _2-1}(u_n-u)\textrm{d}x\\&\quad \le \Vert \xi _1\Vert _{\frac{2}{2-\delta _1}}\left( \Vert u_n\Vert _{2}^{\delta _1-1}+\Vert u\Vert _{2}^{\delta _1-1}\right) \Vert u_n-u\Vert _{2}\\&\qquad +\Vert \xi _2\Vert _{\frac{2}{2-\delta _2}}\left( \Vert u_n\Vert _{2}^{\delta _2-1}+\Vert u\Vert _{2}^{\delta _2-1}\right) \Vert u_n-u\Vert _{2}\\&\quad \le (M_1+M_2)\Vert u_n-u\Vert _{2}=o_n(1), \end{aligned} \end{aligned}$$

(3.15)

where $M_i= \Vert \xi _i\Vert _{\frac{2}{2-\delta _i}}\left( \eta _{2}^{\delta _i-1}M^{\delta _i-1}+\Vert u\Vert _{2}^{\delta _i-1}\right) ,\, i=1,2$. Then, combining (2.3), (3.1), (3.14) and (3.15), for $n\in \mathbb {N}$ large enough, we have

$$\begin{aligned} \begin{aligned} o_n(1)&=\langle I_\lambda '(u_n)-I_\lambda '(u),u_n-u\rangle \\&= \Vert u_n-u\Vert ^2-\int _{\mathbb {R}^N}\left[ f(x,u_n)-f(x,u)\right] (u_n-u) \textrm{d}x\\&\quad -\lambda \int _{\mathbb {R}^N}\left[ g(x,u_n)-g(x,u)\right] (u_n-u) \textrm{d}x\\&=\Vert u_n-u\Vert ^2+o_n(1). \end{aligned} \end{aligned}$$

Consequently, $u_n\rightarrow u$ strongly in H as $n\rightarrow \infty$. Thus, the functional I satisfies the (PS)$_c$ condition for any $c>0$. The proof is completed. $\square$

Let $(X,\Vert \cdot \Vert )$ be a Banach space such that $X=\overline{\oplus _{i=1}^{\infty } X_{i}}$ with ${\text {dim}} X_{i}<+\infty$ for each $i \in \mathbb {N} .$ Set

$$\begin{aligned} Y_{k}=\bigoplus _{i=1}^{k} X_{i}, \quad Z_{k}=\bigoplus _{i=k}^{\infty } X_{i}. \end{aligned}$$

In order to prove Theorem 2.1, we shall use the following Fountain Theorem.

Theorem 3.1

[24, Theorem 3.6] Let X be an infinite dimensional Banach space. Assume that $\varphi \in C^{1}(X,\mathbb {R}), \varphi (-u)=\varphi (u)$ for all $u\in X$. If, for every $k\in \mathbb {N}$, there exist $\rho _k>r_k>0$ such that

$(A_1)$:

$\varphi$ satisfies the (PS)$_{c}$ condition for every $c>0$;

$(A_2)$:

$a_k:=\max \limits _{u\in Y_k,\Vert u\Vert =\rho _k}\varphi (u)\le 0$.

$(A_3)$:

$b_k:=\inf \limits _{u\in Z_k,\Vert u\Vert =r_k}\varphi (u)\rightarrow +\infty$ as $k\rightarrow \infty$.

Then $\varphi$ has a sequence of critical points $\{u_k\}$ such that $\varphi (u_k)\rightarrow +\infty$.

Since $H \hookrightarrow L^{2}\left( \mathbb {R}^{N}\right)$ is compact under assumptions (V) and $L^{2}\left( \mathbb {R}^{N}\right)$ is a separable Hilbert space, then H possesses is a countable orthonormal basis $\{e_{j}\}_{j=1}^{\infty }$. Define

$$\begin{aligned} X_{j}=\mathbb {R}e_{j},\quad Y_{k}=\bigoplus _{j=1}^{k}X_{j},\quad Z_{k}=\overline{\bigoplus _{j=k+1}^{\infty }X_{j}},\quad k\in \mathbb {N}^*. \end{aligned}$$

Then, $H=\overline{\bigoplus _{j=1}^{\infty }X_{j}}$ and $Y_{k}$ is finite dimensional.

Lemma 3.2

Assume that $(V), (F_1)$ and $(g_1)$ hold, then there exist ${\overline{\lambda }}> 0$ and $r_k>0$ such that

$$\begin{aligned} \inf \limits _{u\in Z_k,\Vert u\Vert =r_k}I_\lambda (u)\rightarrow +\infty \quad \text {as } k\rightarrow \infty \end{aligned}$$

whenever $|\lambda |\le {\overline{\lambda }}$.

Proof

Similar to Lemma 3.8 in [24], for any $2\le p< 2_s^*$, we have

$$\begin{aligned} \beta _{k}(p):=\sup \limits _{u\in Z_{k},\Vert u\Vert =1}\Vert u\Vert _{p}\rightarrow 0, \end{aligned}$$

(3.16)

as $k\rightarrow \infty$.

By (2.2), (3.2), (3.9) and (3.16) we obtain

$$\begin{aligned} \begin{aligned} I_\lambda (u)&=\frac{1}{2}\Vert u\Vert ^{2}-\int _{\mathbb {R}^{N}}F(x,u) \textrm{d}x-\lambda \int _{\mathbb {R}^N} G(x,u)\\&\ge \frac{1}{2}\Vert u\Vert ^{2}-\frac{c_1}{2} \Vert u\Vert _2^2-\frac{c_2}{p} \Vert u\Vert _p^{p}-\lambda \left( \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}\Vert u\Vert ^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\Vert u\Vert ^{\delta _2}\right) \\&\ge \frac{1}{2}\Vert u\Vert ^{2}-\frac{c_1}{2} \beta _k^2(2)\Vert u\Vert ^2-\frac{c_2}{p} \beta _k^p(p)\Vert u\Vert ^{p}-|\lambda |\\&\quad \left( \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}\Vert u\Vert ^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\Vert u\Vert ^{\delta _2}\right) . \end{aligned} \end{aligned}$$

According to (3.16), we can choose a large $k_0>1$ so that

$$\begin{aligned} \beta _k^2(2) \le \frac{1}{2c_1},\quad \forall k\ge k_0. \end{aligned}$$

This provides

$$\begin{aligned} I_\lambda (u)\ge \frac{1}{4}\Vert u\Vert ^{2}-\frac{c_2}{p} \beta _k^p(p)\Vert u\Vert ^{p}-|\lambda |\left( \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}\Vert u\Vert ^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\Vert u\Vert ^{\delta _2}\right) . \end{aligned}$$

For any $u\in Z_{k}$ satisfying $\Vert u\Vert \ge 1$, we have

$$\begin{aligned} \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}\Vert u\Vert ^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\Vert u\Vert ^{\delta _2}\le \left( \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\right) \Vert u\Vert ^{\delta _2}, \end{aligned}$$

since $1<\delta _1<\delta _2<2$. Hence, we obtain

$$\begin{aligned} \begin{aligned} I_\lambda (u)&\ge \frac{1}{4}\Vert u\Vert ^{2}-\frac{c_2}{p} \beta _k^p(p)\Vert u\Vert ^{p}-|\lambda |\left( \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}\Vert u\Vert ^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\Vert u\Vert ^{\delta _2}\right) \\&\ge \frac{1}{4}\Vert u\Vert ^{2}-\frac{c_2}{p} \beta _k^p(p)\Vert u\Vert ^{p}-|\lambda |\left( \Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}\right) \Vert u\Vert ^{\delta _2}\\&= \Vert u\Vert ^{\delta _2} \left[ \frac{1}{4}\Vert u\Vert ^{2-\delta _2}-\frac{c_2}{p} \beta _k^p(p)\Vert u\Vert ^{p-\delta _2}-|\lambda |K\right] , \end{aligned} \end{aligned}$$

(3.17)

where $K=\Vert \xi _1\Vert _{\theta _1}\eta _2^{\delta _1}+ \Vert \xi _2\Vert _{\theta _2}\eta _2^{\delta _2}$. For each $k\in \mathbb {N}$ sufficiently large, taking

$$\begin{aligned} r_k:=\left( \frac{p}{8c_2 \beta _{k}^{p}(p)}\right) ^{1/(p-2)}. \end{aligned}$$

Then, by virtue of (3.16) we obtain

$$\begin{aligned} r_k\rightarrow +\infty \quad \text {as}\quad k\rightarrow \infty . \end{aligned}$$

Then, there exists $k_1>1$ such that $r_k\ge 1$ when $k\ge k_1$. By (3.17), for $u\in Z_k, \Vert u\Vert =r_k$, we have

$$\begin{aligned} \begin{aligned} I_\lambda (u)&\ge r_k^{\delta _2}\left( \frac{1}{4} r_k^{2-\delta _2}-\frac{c_2}{p} \beta _k^p(p)r_k^{p-\delta _2}-|\lambda |K \right) \\&=r_k^{\delta _2}\left( \frac{1}{4} r_k^{2-\delta _2}-\frac{c_2}{p} \beta _k^p(p)r_k^{p-\delta _2}-|\lambda |K \right) \\&=r_k^{\delta _2}\left( \frac{1}{8} r_k^{2-\delta _2}-|\lambda |K \right) . \end{aligned} \end{aligned}$$

(3.18)

Putting $\displaystyle \lambda _k=\frac{r_k^{2-\delta _2}}{16 K}$, then, $\lambda _k>0$ and $\lambda _k \rightarrow \infty$ as $k\rightarrow \infty$. Let

$$\begin{aligned} {\overline{\lambda }} = \inf \limits _{k\ge {\overline{k}}}\lambda _k \end{aligned}$$

where ${\overline{k}}=\max \{k_0,k_1\}$, therefore, for any $\lambda \in \mathbb {R}$ satisfying $|\lambda |\le {\overline{\lambda }}$ we get from (3.18)

$$\begin{aligned} I_\lambda (u) \ge \frac{r_k^2}{16}, \quad u\in Z_k,\, \Vert u\Vert =r_k . \end{aligned}$$

Hence, for $k\ge {\overline{k}}$ we deduce

$$\begin{aligned} \inf \limits _{u\in Z_k,\Vert u\Vert =r_k}I_\lambda (u)\ge \frac{r_k^{2}}{16} \rightarrow +\infty \quad \text {as}\quad k\rightarrow \infty \end{aligned}$$

whenever $|\lambda |\le {\overline{\lambda }}$. This completes the proof. $\square$

Lemma 3.3

For any finite dimensional subspace $Y_k \subset H$, there holds

$$\begin{aligned} \max \limits _{u\in Y_k,\Vert u\Vert =\rho _k}I_\lambda (u)\le 0. \end{aligned}$$

Proof

Let $Y_k$ be any finite dimensional subspace of H, we claim that there exists a constant $R_k=R(Y_k)>0$ such that $I_\lambda (u) \le 0$ $\Vert u\Vert \ge R_k$. Otherwise, there is a sequence $\{u_n\}\subset Y_k$ such that

$$\begin{aligned} \Vert u_n\Vert \rightarrow \infty \quad \text { and } \quad I_\lambda (u_n) \ge 0. \end{aligned}$$

(3.19)

Set $v_n = \frac{u_n}{\Vert u_n\Vert }$, then $\Vert v_n\Vert =1$. Therefore, by the Sobolev embedding theorem, up to a subsequence, we can assume $v_n \rightharpoonup v$ in $H, v_n \rightarrow v$ in $L^p(\mathbb {R}^N)$ ($2\le p<2_s^*$) and $v_n \rightarrow v$ a.e. in $\mathbb {R}^N$. Set $E=\{x\in \mathbb {R}^N \, :\, v(x)\ne 0\}$. Since on the finite dimensional subspace $Y_k$ all norms are equivalent, there exists a constant $\alpha _k>0$ such that

$$\begin{aligned} \Vert u\Vert _p \ge \alpha _k\Vert u\Vert ,\quad \forall u\in Y_k, \end{aligned}$$

and then

$$\begin{aligned} \int _{\mathbb {R}^N}\left| u_{n}\right| ^{p} \textrm{d}x \ge \alpha _k^{p}\left\| u_{n}\right\| ^{p}, \quad \forall n \in \mathbb {N}, \end{aligned}$$

which yields

$$\begin{aligned} \alpha _k^p\le \lim _{n\rightarrow \infty }\int _{\mathbb {R}^{N}}\left| v_{n}\right| ^{p} \textrm{d}x=\Vert v_n\Vert _p^p. \end{aligned}$$

Hence ${{\,\textrm{meas}\,}}(E)>0$and then $|u_n(x)|\rightarrow \infty$ for all $x \in E$. Using (2.2) and (3.19) we obtain

$$\begin{aligned} \frac{1}{2}\Vert u_n\Vert ^2\ge \int _{\mathbb {R}^N}F(x,u_n)\textrm{d}x+\lambda \int _{\mathbb {R}^N}G(x,u_n)\textrm{d}x,\quad \forall x\in \mathbb {R}^N. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{1}{2} \ge \int _{\mathbb {R}^N} \frac{F(x,u_n)}{\Vert u_n\Vert ^2} \textrm{d}x+\lambda \int _{\mathbb {R}^N} \frac{G(x,u_n)}{\Vert u_n\Vert ^2} \textrm{d}x. \end{aligned}$$

Then, by (3.10) and Fatou’s Lemma we deduce

$$\begin{aligned}{} & {} \frac{1}{2} \ge \liminf _{n\rightarrow \infty } \int _{\mathbb {R}^N} \frac{F(x,u_n)}{|u_n|^2}|v_n|^2 \textrm{d}x \\{} & {} \quad \ge \liminf _{n\rightarrow \infty } \int _{E} \frac{F(x,u_n)}{|u_n|^2}|v_n|^2 \textrm{d}x\ge \int _{E}\liminf _{n\rightarrow \infty } \frac{F(x,u_n)}{|u_n|^2}|v_n|^2 \textrm{d}x=+\infty . \end{aligned}$$

We have a contradiction. This shows that there exists a constant $R_k=R(Y_k)>0$ such that $I(u) \le 0$ for all $u\in Y_k\setminus B_{R_k}(0)$. Hence, choosing $\rho _k>\max \{R_k,r_k\}$, we conclude that

$$\begin{aligned} \max \limits _{u\in Y_k,\Vert u\Vert =\rho _k} I_\lambda (u) \le 0. \end{aligned}$$

$\square$

Proof of Theorem 2.1

We have $I_\lambda \in C^1(H,\mathbb {R})$ is even in view of $(F_4)$ and $(g_2)$. On the other hand, by Lemmas 3.1 and 3.3, the functional $I_\lambda$ satisfies the conditions $(A_1)$–$(A_2)$ of the Fountain Theorem 3.1, respectively. Moreover, condition $(A_3)$ is satisfied whenever $|\lambda |\le {\overline{\lambda }}$ due to Lemma 3.2. Thus, the functional $I_\lambda$ has a sequence of critical points $\{u_k\}\subset H$ such that $I_\lambda (u_k) \rightarrow \infty$ as $k \rightarrow \infty$, whenever $|\lambda |\le {\overline{\lambda }}$, that is, Eq. (1.5) possesses infinitely many solutions. $\square$