Infinitely Many Small Energy Solutions to Nonlinear Kirchhoff–Schrödinger Equations with the p-Laplacian

Abstract

This paper is devoted to deriving the multiplicity result of solutions to the nonlinear elliptic equations of Kirchhoff–Schrödinger type on a class of a nonlocal Kirchhoff coefficient which slightly differs from the previous related works. More precisely, the main purpose of this paper, under the various conditions for a nonlinear term, is to show that our problem has a sequence of infinitely many small energy solutions. In order to obtain such a multiplicity result, the dual fountain theorem is used as the primary tool.

1 Introduction

The present paper is devoted to the multiplicity result of nontrivial weak solutions to the p-Laplacian equations of Kirchhoff–Schrödinger type:

where $1<p<N$, $h:{\mathbb {R}^{N}}\times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function and $\mathfrak {P}:\mathbb {R}^{N} \rightarrow (0,\infty )$ is a potential function with

• (V) $\mathfrak {P} \in C(\mathbb {R}^{N})$, $\mathop {\mathrm {ess\,inf}}\limits _{x\in \mathbb {R}^N} \mathfrak {P}(x)> 0$, and meas$\left\{ x\in \mathbb {R}^{N}:\mathfrak {P}(x)\le P_0\right\} <+\infty$ for all $P_0\in \mathbb {R}$.

Also the Kirchhoff function $M:[0,\infty ) \rightarrow \mathbb {R}^+$ fulfils the conditions as follows:

• (K1) $M\in C(\mathbb {R}_{0}^{+})$ fulfils $\inf _{t\in \mathbb {R}^{+}} M(t) \ge m_0 > 0$, where $m_0$ is a constant.

• (K2) There exist a constant $\vartheta \ge 1$ and a nonnegative constant K such that $\vartheta \mathcal {M}(t)=\vartheta \int _0^t M(\tau )d \tau \ge M(t)t$ and

  $$\begin{aligned} \widehat{\mathcal {M}}(st)\le \widehat{\mathcal {M}}(t)+K \end{aligned}$$

  for $t \ge 0$ and $s\in [0,1]$, where $\widehat{\mathcal {M}}(t) =\vartheta \mathcal {M}(t)-M(t)t$.

The study on elliptic problems regarding the nonlocal Kirchhoff term was first provided by Kirchhoff [16] to study an extension of the classical D’Alembert’s wave equation by taking into consideration the changes in the length of the strings during the vibrations. The variational Kirchhoff-type problems have a strong background in diverse applications in physics and have been intensively investigated by many researchers in recent years, for instance [3,4,5, 10, 18, 21, 24, 26] and the references therein. The authors in [8] first proposed a detailed discussion about the physical meaning based on the fractional Kirchhoff model. Under some appropriate conditions, they established the existence of nontrivial solutions by utilizing the mountain pass theorem and a truncation argument on nonlocal Kirchhoff term. In this paper, the conditions enforced on the nondegenerate Kirchhoff function $M:\mathbb {R}^{+}_{0}\rightarrow \mathbb {R}^{+}_{0}$ are that M is a continuous function satisfying (K1), see also [25] and references therein. But, the increasing condition excludes the case that is not monotone, for example,

$$\begin{aligned} M(t)=(1+t)^{-1}+(1+t)^{\alpha } \text { with } \alpha \in (0,1) \end{aligned}$$

(1.1)

for all $t \in \mathbb {R}_{0}^{+}.$ For this purpose, Pucci-Xiang-Zhang in [26] provided the multiplicity result of solutions to a class of Schrödinger–Kirchhoff-type equations with fractional p-Laplacian, where the nondegenerate Kirchhoff coefficient M satisfies the condition:

• (K3) For $0<s<1$, there is $\vartheta \in [1,\frac{N}{N-sp})$ such that $\vartheta \mathcal {M}(t)\ge M(t)t$ for any $t\ge 0$.

Very recently, Huang–Deng [11] investigated the existence of a positive ground state solution for Kirchhoff-type problem with critical exponential growth under the assumption:

• (K4) There is $\vartheta > 1$ such that $\frac{M(t)}{t^{\vartheta -1}}$ is nonincreasing for $t > 0$.

From this assumption, it is obvious that $\vartheta \mathcal {M}(t)-M(t)t$ is nondecreasing for all $t \ge 0$ and thus we get the following condition

• (K3)’ there exists $\vartheta > 1$ such that $\vartheta \mathcal {M}(t)\ge M(t)t$ for any $t\ge 0$,

that is weaker than (K4). A classical model for M satisfying (K1) and (K3)$^{\prime }$ is given by $M(t)=1 +a t^\vartheta$ with $a\ge 0$ for all $t\ge 0$. Hence the condition (K3)$^{\prime }$ includes the above typical example as well as the case that is not monotone. In this reason, the nonlinear elliptic problems of Kirchhoff type with (K3)$^{\prime }$ (or (K3)) have been widely studied by many researchers in recent years; see [5, 10, 18, 26]

Remark 1.1

We know that the example (1.1) satisfies the conditions (K3) and (K4) for any $\vartheta =k+1$ with $0<k<1$. However we will consider an example that does not satisfy (K3) and (K4) under suitable conditions $\vartheta$, p and N. Let us consider

$$\begin{aligned} M(t)=\left( 1+\frac{t^{2}}{\sqrt{1+t^{4}}}\right) t+(1+t)^{-\frac{5}{2}} \end{aligned}$$

with its primitive function

$$\begin{aligned} \mathcal {M}(t)=\frac{1}{2}\left( t^{2}+\sqrt{1+t^{4}}-1\right) -\frac{2}{3}(1+t)^{-\frac{3}{2}}+\frac{2}{3} \end{aligned}$$

for all $t\ge 0$. Then it is clear that M is not monotone and

$$\begin{aligned} \widehat{\mathcal {M}}(t)&=\vartheta \mathcal {M}(t)-M(t)t\\&=\left( \frac{\vartheta }{2}-1\right) t^{2}+\frac{\vartheta }{2}\sqrt{1+t^{4}}-t(1+t)^{-\frac{5}{2}}-\frac{2\vartheta }{3}(1+t)^{-\frac{3}{2}}-\frac{t^{4}}{\sqrt{1+t^{4}}}+\frac{\vartheta }{6}. \end{aligned}$$

If $p=2$ and $N=4$ in (K3), then we could not be found a constant $\vartheta \in [1,2)$ satisfying $\widehat{\mathcal {M}}(t) \ge 0$ for any $t \ge 0$ by being $\lim _{t\rightarrow \infty }\widehat{\mathcal {M}}(t)=-\infty$. However, if $\vartheta \ge 2$, we get $\widehat{\mathcal {M}}(t)\ge 0$ for any $t\ge 0$. On the other hand, if we set $\vartheta =2$, then one has

$$\begin{aligned} \widehat{\mathcal {M}}(t) =\frac{1}{\sqrt{1+t^{4}}}-t(1+t)^{-\frac{5}{2}}-\frac{4}{3}(1+t)^{-\frac{3}{2}}+\frac{1}{3} \end{aligned}$$

is not nondecreasing for $t\ge 0$ from a straightforward calculation. This means this example does not ensure the condition (K4). This implies that

$$\begin{aligned} \widehat{\mathcal {M}}(t)-\widehat{\mathcal {M}}(st)\ge 0 \end{aligned}$$

is not satisfied. However, we can find a nonnegative constant K that satisfies our condition (K2). Of course, because (K4) implies the condition (K2), our condition is a generalization of the condition (K4).

Motivated by the fact illustrated in the remark above, the main purpose of the present paper is devoted to deriving the existence result of multiple solutions to the problem of Kirchhoff–Schrödinger type involving the p-Laplacian on a class of a nonlocal Kirchhoff coefficient M which is slightly different from them as well as the cases considered the previous related works. To do this, we assume that h satisfies the assumptions as follows:

(${\Psi }1$):

$h: \mathbb {R}^N\times \mathbb {R} \rightarrow \mathbb {R}$ is the Carathéodory function and there are $b_{1}>0$ and $\gamma \in [p,p^{*}),$ $0\le \sigma _{0}\in L^{\gamma '}(\mathbb {R}^{N})\cap L^{\infty }(\mathbb {R}^{N})$ such that

$$\begin{aligned} \left|h(x,\ell )\right|\le \sigma _{0}(x)+b_{1}\left|\ell \right|^{q-1} \end{aligned}$$

for all $(x,\ell )\in \mathbb {R}^{N}\times \mathbb {R}$, where $1<p<q<p^{*}=\frac{Np}{N-p}$.

(${\Psi }2$):

There is a constant $\mu \ge 1$ such that

$$\begin{aligned} \mu \mathcal {H}(x,\ell ) \ge \mathcal {H}(x,s\ell ) \end{aligned}$$

for $(x,\ell )\in \mathbb {R}^{N}\times \mathbb {R}$ and $s \in [0,1]$. Here $\mathcal {H}(x,\ell )=h(x,\ell )\ell -p\vartheta H(x,\ell )$, where $H(x,\ell )=\int _{0}^{\ell }h(x,s)\,ds$ and $\vartheta$ was given in (K2).

(${\Psi }3$):

There are positive constants $\mu >p\vartheta$ and $T>0$ such that

$$\begin{aligned} \mu H(x,\ell )\le \ell h(x,\ell ) \end{aligned}$$

for all $x\in \mathbb {R}^N$ and $|\ell |\ge T$.

(${\Psi }4$):

There exist $C>0$, $1<\kappa <p$, $\tau >1$ with $p\le \tau ^\prime \kappa \le p^{*}$ and a positive function $\eta \in L^{\tau }(\mathbb {R}^{N})\cap L^{\infty }(\mathbb {R}^{N})$ such that

$$\begin{aligned} \liminf _{\left|\ell \right|\rightarrow 0}\frac{h(x,\ell )}{\eta (x)\left|\ell \right|^{\kappa -2}\ell }\ge C \end{aligned}$$

uniformly for almost all $x\in \mathbb {R}^{N}$.

The condition ($\Psi 2$) is initially suggested by the work of Jeanjean [13]. In particular, Liu [22] obtained the existence result of a ground state to the superlinear p-Laplacian problems:

$$\begin{aligned} -\text {div}(|\nabla v|^{p-2}\nabla v) + \mathfrak {P}(x)|v|^{p-2}v =h(x,v)\ \text {in} \,\,\mathbb {R}^N. \end{aligned}$$

(1.2)

Here the potential function $\mathfrak {P}\in C({\mathbb {R}}^N)$ satisfies appropriate conditions and the Carathéodory function $h:\mathbb {R}^{N}\times \mathbb {R} \rightarrow \mathbb {R}$ fulfils the conditions ($\Psi 1$)–($\Psi 2$) and the following condition:

(${\Psi }5$):

$H(x,\ell )=o(|\ell |^{p})$ as $\ell \rightarrow 0$ uniformly for almost all $x\in {\mathbb {R}}^N$.

In this direction, there have been substantial researches dealing with the p-Laplacian problem; see [22, 23] and see also [1, 18, 28] for variable exponents $p(\cdot )$. However, because a nonlocal Kirchhoff coefficient M exists, we cannot achieve the same results even if we follow similar methods as in [1, 18, 22, 23, 28]. More precisely, under the assumptions ($\Psi 1$)–($\Psi 2$) and ($\Psi 5$), we cannot guarantee the Palais–Smale type compactness condition for the energy function when the Kirchhoff function M satisfies the condition (K3)$^{\prime }$. Particularly, it is essential that $\widehat{\mathcal {M}}(t)$ is nondecreasing for all $t \ge 0$ to verify this compactness condition of an energy functional corresponding to elliptic problems with the nonlinear term satisfying ($\Psi 2$). For this reason, when (K3)$^{\prime }$ was satisfied, many authors considered a condition of the nonlinear terms different from ($\Psi 2$); see [2, 5, 7, 10, 15, 18, 26, 27]. From this perspective, one of the novelties of the present paper is to obtain the existence result of a sequence of infinitely many energy solutions to $(P_{\lambda })$ without the monotonicity of $\widehat{\mathcal {M}}(t)$ when ($\Psi 2$) is assumed.

When (V) is satisfied, Lin-Tang [20] has recently investigated the various multiplicity results of solutions to the problem (1.2) with mild conditions for the superlinear term h which differ from ($\Psi 2$). Inspired by this work, the author in [15] have derived several existence results on infinitely many solutions to Kirchhoff–Schrödinger equations involving the $p(\cdot )$-Laplace-type operator for the case that the Kirchhoff coefficient fulfils (K3)$^{\prime }$ and the nonlinear term holds ($\Psi 3$). One of the key ingredients to get these results is the fact that the potential function $\mathfrak {P}\in C(\mathbb {R}^N,(0,\infty ))$ is coercive, that is, $\lim _{\left|x\right|\rightarrow \infty }\mathfrak {P}(x)=+\infty$ which is crucial to ensure the compactness condition of the Palais–Smale type. However, in order to prove this condition, we employ a weaker condition (V)  than the coercivity of the function $\mathfrak {P}$. Furthermore, another novelty of this paper is to discuss our main consequence without assuming the condition ($\Psi 5$) which plays a decisive role in showing the compactness condition of the Palais–Smale type and guaranteeing assumptions in the dual fountain theorem. Let us consider the function

$$\begin{aligned} h(x,\ell )=\sigma (x)\left( \eta (x)\left|\ell \right|^{\kappa -2}s+\left|\ell \right|^{q-2}\ell \ln {\left( 1+\left|\ell \right|\right) }+\frac{\left|\ell \right|^{q-1}\ell }{1+\left|\ell \right|}\right) \end{aligned}$$

with its primitive function

$$\begin{aligned} H(x,\ell )=\sigma (x)\left( \frac{\eta (x)}{\kappa }\left|\ell \right|^{\kappa }+\frac{1}{q}\left|\ell \right|^{q}\ln {\left( 1+\left|\ell \right|\right) }\right) \end{aligned}$$

for all $\ell \in \mathbb {R}$, where $p<q$ and $\sigma \in C(\mathbb {R}^{N},{\mathbb {R}})$ with $0<\inf _{x\in {\mathbb {R}}^N}\sigma (x)\le \sup _{x\in {\mathbb {R}}^N}\sigma (x)<\infty$. Then this example fulfils the assumptions ($\Psi 1$)–($\Psi 2$) and ($\Psi 4$), but not ($\Psi 3$) and ($\Psi 5$). On the other hand, we give an example that holds the condition ($\Psi 3$), but not ($\Psi 2$).

Example 1.2

Let us consider the function

$$\begin{aligned} h(x,\ell )=\eta (x)\left|\ell \right|^{\kappa -2}\ell +\sigma (x)|\ell |^{p\vartheta -1}\ell \Big [(p\vartheta +3)\ell ^{2}-2(p\vartheta +2)|\ell |+(p\vartheta +1)\Big ] \end{aligned}$$

with its primitive function

$$\begin{aligned} H(x,\ell )=\frac{\eta (x)}{\kappa }|\ell |^{\kappa }+\sigma (x)(|\ell |^{p\vartheta +3}-2|\ell |^{p\vartheta +2}+|\ell |^{p\vartheta +1}), \end{aligned}$$

where $\eta$ is given in ($\Psi 4$) and $\sigma \in C(\mathbb {R}^{N},{\mathbb {R}})$ with $0<\inf _{x\in {\mathbb {R}}^N}\sigma (x)\le \sup _{x\in {\mathbb {R}}^N}\sigma (x)<\infty$. Set $\tilde{\omega }:=\inf _{x\in {\mathbb {R}}^N}\sigma (x)$ for all $x\in {\mathbb {R}}^N$, then we have

$$\begin{aligned}&h(x,\ell )\ell -\mu H(x,\ell ) \ge |\!|\eta |\!|_{L^{\infty }(\mathbb {R}^N)}\Big (1-\mu \Big )|\ell |^{p\vartheta +1}\\&\quad +\sigma (x)\Big ((p\vartheta +3-\mu )|\ell |^{p\vartheta +3}-2(p\vartheta +2-\mu )|\ell |^{p\vartheta +2}+(p\vartheta +1-\mu )|\ell |^{p\vartheta +1}\Big )\\&\ge |\!|\eta |\!|_{L^{\infty }(\mathbb {R}^N)}\Big (1-\mu \Big )|\ell |^{p\vartheta +1}\\&\quad +\tilde{\omega }\Big ((p\vartheta +3-\mu )|\ell |^{2}-2(p\vartheta +2-\mu )|\ell |-(\mu -p\vartheta -1)\Big )|\ell |^{p\vartheta +1}\\&\ge \min \{|\!|\eta |\!|_{L^{\infty }(\mathbb {R}^N)},\tilde{\omega }\}\Big (|\ell |^{2}+(p\vartheta +2-\mu )(|\ell |^{2}-2|\ell |)-(2\mu -p\vartheta -2)\Big )|\ell |^{p\vartheta +1}\\&\ge 0 \end{aligned}$$

for $|\ell |\ge \ell _{0}$, where $\ell _{0}>2$ is chosen such that $\ell _{0}^2-2\ell _{0}\ge 0$ and $\mu$ is a value that lies in the interval $(p\vartheta ,p\vartheta +2]$. Hence, this example satisfies the condition ($\Psi 3$), but not ($\Psi 2$).

To this end, on a class of the Kirchhoff function M and the superlinear term h which differ from the previous related works [2, 5, 7, 10, 15, 18, 26, 27], we provide the existence result of multiple small energy solutions by taking advantage of the dual fountain theorem as the key tool. The fundamental idea of our proof for the existence of a sequence of infinitely many small energy solutions comes from the recent studies [14, 15]. Such existence results for nonlinear elliptic problems are particularly inspired by contributions from recent works [1, 9, 12, 18, 21,22,23, 29], and the references therein.

The outline of this paper is as follows. We present some necessary preliminary knowledge of function spaces which we will use along the paper. Next, we provide the variational framework related to problem $(P_{\lambda })$ and then we obtain the various existence results of multiple small energy solutions to the p-Laplacian equations of Kirchhoff–Schrödinger type under conditions ($\Psi 1$)–($\Psi 4$).

2 Main Results

In this section, we present the existence of infinitely many nontrivial solutions to problem $(P_{\lambda })$, by making use of the dual fountain theorem as the primary tool.

When the potential function $\mathfrak {P}$ satisfies (V), let us define the linear subspace

$$\begin{aligned} X=\left\{ v\in W^{1,p}(\mathbb {R}^{N}): \int _{\mathbb {R}^{N}}\left( \left|\nabla v\right|^{p}+\mathfrak {P}(x)\left|v\right|^{p}\right) \,\textrm{d}x < +\infty \right\} \end{aligned}$$

with the norm

$$\begin{aligned} |\!|v|\!|_{X}=\left( \int _{\mathbb {R}^{N}}\left( \left|\nabla v\right|^{p}+\mathfrak {P}(x)\left|v\right|^{p}\right) \,\textrm{d}x\right) ^{\frac{1}{p}}, \end{aligned}$$

which is equivalent to the norm $|\!|v|\!|_{W^{1,p}(\mathbb {R}^{N})}$.

Lemma 2.1

([1]) If the assumption (V) is satisfied, then

• (1) we have a compact embedding $X\hookrightarrow L^{p}(\mathbb {R}^{N})$;

• (2) there exists a compact embedding $X\hookrightarrow L^{q}(\mathbb {R}^{N})$ for any q with $p<q<p^*$.

Definition 2.2

By a solution of problem $(P_{\lambda })$, we mean a function $v\in X$ such that

$$\begin{aligned} M\left( \int _{{\mathbb {R}}^N} \left|\nabla v\right|^{p}\,\textrm{d}x\right)&\int _{\mathbb {R}^{N}} \left|\nabla v\right|^{p-2}\nabla v \cdot \nabla z \,\textrm{d}x+\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v\right|^{p-2}vz\,\textrm{d}x\\&=\int _{\mathbb {R}^{N}}h(x,v) z\,\textrm{d}x \end{aligned}$$

for all $z \in X$.

Let us define the functional $\Phi : X \rightarrow \mathbb {R}$ by

$$\begin{aligned} \Phi (v)=\frac{1}{p} \mathcal {M}\left( \int _{{\mathbb {R}}^N} \left|\nabla v\right|^{p}\,\textrm{d}x\right) +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v\right|^{p}\,\textrm{d}x.\end{aligned}$$

Then it is not difficult to prove that $\Phi$ is well defined on X, $\Phi \in C^{1}(X,\mathbb {R})$ and its Fréchet derivative is defined as

$$\begin{aligned} \langle {\Phi ^{\prime }(v),z}\rangle =M\left( \int _{{\mathbb {R}}^N} \left|\nabla v\right|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}{\left|\nabla v\right|^{p-2}\nabla v\cdot \nabla z}\,\textrm{d}x+\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v\right|^{p-2}vz\,\textrm{d}x, \end{aligned}$$

where $\left\langle \cdot ,\cdot \right\rangle$ denotes the pairing of X and its dual $X^{*}$.

According to the similar arguments in [17, 19], the following consequence is easily ensured, so we omit the proof.

Lemma 2.3

Assume that (K1)–(K2) and (V) hold. The functional $\Phi :X\rightarrow \mathbb {R}$ is weakly lower semicontinuous and convex on X. In addition, $\Phi ^{\prime }$ is a mapping of type $(S_+)$, i.e., if $v_{n}\rightharpoonup v$ in X and $\lim \sup _{n\rightarrow \infty }\left\langle \Phi ^\prime (v_{n})-\Phi ^\prime (v), v_{n}-v\right\rangle \le 0$, then $v_{n}\rightarrow v$ in X as $n\rightarrow \infty$.

Define the functional $\Psi :X\rightarrow \mathbb {R}$ by

$$\begin{aligned} \Psi (v)=\int _{\mathbb {R}^N}H(x,v)\,\textrm{d}x. \end{aligned}$$

Then $\Psi \in C^{1}(X,\mathbb {R})$ and its Fréchet derivative is

$$\begin{aligned} \left\langle \Psi ^{\prime }(v),z\right\rangle \,=\int _{\mathbb {R}^N}h(x,v)z\,\textrm{d}x \end{aligned}$$

for any $v,z \in X$. Next the functional $I_{\lambda }:X\rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} I_{\lambda }(v)=\Phi (v)-\lambda \Psi (v). \end{aligned}$$

Then it is clear that $I_{\lambda }\in C^{1}(X,\mathbb {R}^{N})$ and its Fréchet derivative is

$$\begin{aligned} \left\langle I_{\lambda }^{\prime }(v),z\right\rangle&=M\left( \int _{{\mathbb {R}}^N} \left|\nabla v\right|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}\left|\nabla v\right|^{p-2}\nabla v\cdot \nabla z\,\textrm{d}x+\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v\right|^{p-2}vz\,\textrm{d}x\\&\qquad -\lambda \int _{\mathbb {R}^{N}}h(x,v)z\,\textrm{d}x \end{aligned}$$

for any $v,z\in X$.

The following assertion follows from proceeding the similar arguments as in [18, Lemma 3.4].

Lemma 2.4

Assume that (V) and ($\Psi 1$) is satisfied. Then the functionals $\Psi$ and $\Psi ^{\prime }$ are compact operators on X.

With the help of Lemmas 2.3 and 2.4, we derive that the energy functional $I_{\lambda }$ ensures the Cerami condition at level c ($(C)_c$-condition, for short), i.e., any sequence $\left\{ y_{n}\right\} \subset X$ such that $I_{\lambda }(y_{n})\rightarrow c$ and $|\!|I_{\lambda }^{\prime }(y_{n})|\!|_{X^{*}}(1+|\!|y_{n}|\!|_{X})\rightarrow 0$ as $n\rightarrow \infty$ has a convergent subsequence. This is essential in establishing the existence of nontrivial weak solutions for our problem. The basic idea of proofs of this assertion follows similar arguments as in [12, 18].

Lemma 2.5

Let (K1)–(K2), (V) and ($\Psi 1$)–($\Psi 2$) hold. Furthermore, we assume that

(${\Psi }6$):

$H(x,\ell )\ge 0$ for all $(x,\ell )\in {\mathbb {R}}^{N}\times {\mathbb {R}}$ and $\lim _{\left|\ell \right|\rightarrow \infty }{\frac{H(x,\ell )}{\left|\ell \right|^{\vartheta p}}}=\infty$ uniformly for almost all $x\in \mathbb {R}^{N}$.

Then the functional $I_{\lambda }$ fulfils the $(C)_c$-condition for any $\lambda >0$.

Proof

For any $c \in \mathbb {R}$, let $\{v_{n}\}$ be a $(C)_c$-sequence in X, i.e.,

$$\begin{aligned} I_{{\lambda }}(v_{n})\rightarrow c \ \text {and} \ \left\langle I^{\prime }_{\lambda }(v_{n}),v_{n}\right\rangle =o(1) \rightarrow 0, \ \text {as} \ n\rightarrow \infty . \end{aligned}$$

(2.1)

By virtue of Lemmas 2.3 and 2.4, we have that the functional $\Phi ^{\prime }$ is mapping of type $(S_{+})$ and $\Psi ^{\prime }$ is compact. Thus, because $I_{\lambda }^{\prime }$ is of type $(S_{+})$ and X is reflexive, it is enough to assure the boundedness of the sequence $\{v_{n}\}$ in X. To this end, arguing by contradiction, suppose that $|\!|v_{n}|\!|_{X}>1$ and $|\!|v_{n}|\!|_{X}\rightarrow \infty$ as $n\rightarrow \infty$, and define a sequence $\left\{ \varpi _{n}\right\}$ by $\varpi _{n}={v_{n}}/{|\!|v_{n}|\!|_{X}}$. Then, up to a subsequence, still denoted by $\left\{ \varpi _{n}\right\}$, we infer $\varpi _{n}\rightharpoonup \varpi$ in X as $n\rightarrow \infty$ and, according to Lemma 2.1,

$$\begin{aligned}&\varpi _{n}(x) \rightarrow \varpi (x) \text { a.e. in }\mathbb {R}^{N}, \quad \varpi _{n} \rightarrow \varpi \text { in }L^{q}(\mathbb {R}^{N}) \nonumber \\&\text {and } \quad \varpi _{n} \rightarrow \varpi \text { in }L^{p}(\mathbb {R}^{N}) \quad \text { as } \quad n\rightarrow \infty . \end{aligned}$$

(2.2)

Due to (K1)–(K2), and the relation (2.1), we have

$$\begin{aligned} I_{\lambda }(v_{n})&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\,\textrm{d}x\right) +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x-\lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x\nonumber \\&\ge \frac{1}{p\vartheta }M\left( \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\,\textrm{d}x\nonumber \\&\qquad \qquad +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x-\lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x\nonumber \\&\ge \frac{m_0}{p\vartheta }\int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\textrm{d}x + \frac{1}{p}\int _{\mathbb {R}^N}\mathfrak {P}(x)|v_{n}|^{p}\textrm{d}x -\lambda \int _{\mathbb {R}^{N}}H(x,v_{n})\textrm{d}x\nonumber \\&\ge \frac{\min \left\{ m_{0},\vartheta \right\} }{\vartheta p}|\!|v_{n}|\!|_{X}^{p} -\lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x +o(1). \end{aligned}$$

(2.3)

Since $|\!|v_{n}|\!|_{X}\rightarrow \infty$ as $n\rightarrow \infty$, we assert by (2.3) that

$$\begin{aligned} \lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x \ge \frac{\min \left\{ m_0,\vartheta \right\} }{\vartheta p}|\!|v_{n}|\!|_{X}^{p}- I_\lambda (v_{n})\rightarrow \infty \quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

(2.4)

In addition, the assumption (K2) implies that

$$\begin{aligned} I_{\lambda }(v_{n})&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\textrm{d}x\right) +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\textrm{d}x -\lambda \int _{\mathbb {R}^{N}}H(x,v_{n})\textrm{d}x\nonumber \\&\le \frac{1}{p}\mathcal {M}(1)\left( 1+\left( \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\textrm{d}x\right) ^{\vartheta }\right) \nonumber \\&\qquad \qquad +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\textrm{d}x-\lambda \int _{\mathbb {R}^{N}}H(x,v_{n})\textrm{d}x\nonumber \\&\le \max \{\mathcal {M}(1),\frac{1}{p}\}\left( 1+\int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\textrm{d}x+\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\textrm{d}x\right) ^{\vartheta }\nonumber \\&\qquad \qquad -\lambda \int _{\mathbb {R}^{N}}H(x,v_{n})\textrm{d}x\nonumber \\&\le 2^{\vartheta }\max \{\mathcal {M}(1),\frac{1}{p}\}|\!|v_{n}|\!|^{\vartheta p}_{X}-\lambda \int _{\mathbb {R}^{N}}H(x,v_{n})\textrm{d}x, \end{aligned}$$

(2.5)

where $\mathcal {M} (\varsigma )\le \mathcal {M}(1)\left( 1+{\varsigma }^{\vartheta }\right)$ for all $\varsigma \in \mathbb {R}^+$ because if $0\le \varsigma <1$, then $\mathcal {M} (\varsigma )=\int _{0}^\varsigma M(\ell )\,d\ell \le \mathcal {M}(1)$, and if $\varsigma >1$, then $\mathcal {M} (\varsigma )\le \mathcal {M}(1)\varsigma ^{\vartheta }$. Then we obtain by the relation (2.5) that

$$\begin{aligned} 2^{\vartheta }\max \{\mathcal {M}(1),\frac{1}{p}\} \ge \frac{1}{|\!|v_{n}|\!|_{X}^{\vartheta p}}\left( \lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x+I_\lambda (v_{n})\right) . \end{aligned}$$

(2.6)

Set $\Lambda _{1}=\left\{ x\in \mathbb {R}^{N}: \varpi (x)\not =0 \right\}$. Assume that $\text {meas}(\Lambda _1)\ne 0$. By the convergence (2.2), we infer that $\left|v_{n}(x)\right|=\left|\varpi _{n}(x)\right||\!|v_{n}|\!|_{X}\rightarrow \infty$ as $n\rightarrow \infty$ for all $x\in \Lambda _{1}$. Furthermore, owing to ($\Psi 6$) one has

$$\begin{aligned} \lim _{n\rightarrow \infty }{\frac{H(x,v_{n})}{|\!|v_{n}|\!|_{X}^{\vartheta p}}} = \lim _{n\rightarrow \infty }{\frac{H(x,v_{n})}{\left|v_{n}\right|^{\vartheta p}}\left|\varpi _{n}\right|^{\vartheta p} } = \infty \end{aligned}$$

(2.7)

for all $x\in \Lambda _{1}$. In accordance with relations (2.4)–(2.7) and the Fatou lemma, we infer that

$$\begin{aligned} \frac{2^{\vartheta }}{\lambda }\max \{\mathcal {M}(1),p^{-1}\}&={\liminf _{n\rightarrow \infty }{\frac{2^{\vartheta }\max \{\mathcal {M}(1),\frac{1}{p}\}\ {\int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x}}{\lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x+I_{\lambda }(v_{n})}}} \\&\ge {\liminf _{n\rightarrow \infty }\int _{\mathbb {R}^{N}}{\frac{H(x,v_{n})}{|\!|v_{n}|\!|^{\vartheta p}_{X}}}}\,\textrm{d}x\\&\ge \liminf _{n\rightarrow \infty }{\int _{\Lambda _{1}}{\frac{H(x,v_{n})}{|\!|v_{n}|\!|^{\vartheta p}_{X}}}\,\textrm{d}x}\\&\ge \int _{\Lambda _{1}}{\liminf _{n\rightarrow \infty }{\frac{H(x,v_{n})}{|\!|v_{n}|\!|^{\vartheta p}_{X}}}}\,\textrm{d}x\\&= \int _{\Lambda _{1}}{\liminf _{n\rightarrow \infty }{\frac{H(x,v_{n})}{\left|v_{n}\right|^{\vartheta p}}}\left|\varpi _{n}\right|^{\vartheta p}}\,\textrm{d}x=\infty , \end{aligned}$$

which is a contradiction. Hence we have that $\text {meas}(\Lambda _1)=0$ and thus $\varpi (x)=0$ for almost all $x\in \mathbb {R}^{N}$. Since $I_{\lambda }(\tau v_{n})$ is continuous in $\tau \in [0,1]$, for each $n\in \mathbb {N}$, there is an element $\tau _{n}$ in [0, 1] such that

$$\begin{aligned} I_{\lambda }(\tau _{n}v_{n}):= \max _{\tau \in [0,1]}\,{I_{\lambda }(\tau v_{n})}. \end{aligned}$$

Let $\left\{ d_{k}\right\}$ be a positive sequence of real numbers satisfying $\lim _{k\rightarrow \infty }{d_{k}}=\infty$ and $d_{k}>1$ for any k. Then it is immediate that $|\!|d_{k}\varpi _{n}|\!|_{X}=d_{k}>1$ for any k and n. Fix k, since $\varpi _{n}\rightarrow 0$ strongly in the spaces $L^{q}(\mathbb {R}^{N})$ as $n\rightarrow \infty$, it follows from the continuity of Nemytskii operator that $H(x,d_{k}\varpi _{n})\rightarrow 0$ in $L^{1}(\mathbb {R}^{N})$ as $n\rightarrow \infty$. Hence we assert

$$\begin{aligned} \lim _{n\rightarrow \infty }{\int _{\mathbb {R}^{N}}{H(x,d_{k}\varpi _{n})}}\,\textrm{d}x=0. \end{aligned}$$

(2.8)

Since $|\!|v_{n}|\!|_{X}\rightarrow \infty$ as $n\rightarrow \infty$, we have $|\!|v_{n}|\!|_{X}>d_{k}$ for large enough n. Thus we know by (2.8) that

$$\begin{aligned} I_{\lambda }(\tau _{n}v_{n})&\ge I_{\lambda }\left( \frac{d_{k}}{|\!|v_{n}|\!|_{X}}v_{n}\right) =I_{\lambda }(d_{k}\varpi _{n})\\&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^{N}}\left|\nabla d_{k}\varpi _{n}\right|^{p}\textrm{d}x\right) +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|d_{k}\varpi _{n}\right|^{p}\textrm{d}x\\&\qquad \qquad -\lambda \int _{\mathbb {R}^{N}}H(x,d_{k}\varpi _{n})\textrm{d}x\\&\ge \frac{m_{0}}{p\vartheta }\int _{\mathbb {R}^{N}}{\left|\nabla d_{k}\varpi _{n}\right|^{p}}\,\textrm{d}x+\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|d_{k}\varpi _{n}\right|^{p}\,\textrm{d}x\\&\qquad \qquad -\lambda {\int _{\mathbb {R}^{N}}{H(x,d_{k}\varpi _{n})}}\,\textrm{d}x\\&\ge \frac{\min \left\{ m_{0},\vartheta \right\} }{\vartheta p}|\!|d_{k}\varpi _{n}|\!|^{p}_{X}-\int _{\mathbb {R}^{N}}H(x,d_{k}\varpi _{n})\textrm{d}x\\&\ge \frac{\min \left\{ m_{0},\vartheta \right\} }{2\vartheta p}d_{k}^{p} \end{aligned}$$

for n large enough. Then letting n and k tend to infinity, it implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }{I_{\lambda }(\tau _{n}v_{n})}=\infty . \end{aligned}$$

(2.9)

Since $I_{\lambda }(0)=0$ and $I_{\lambda }({v}_{n})\rightarrow c$ as $n\rightarrow \infty$, it is clear that $\tau _{n}\in (0,1)$, and $\left\langle I_{\lambda }^{\prime }(\tau _{n}{v}_{n}),\tau _{n}{v}_{n}\right\rangle =0$. Therefore, by means of the assumption ($\Psi 2$), we deduce that

$$\begin{aligned}&\frac{1}{\mu }I_{\lambda }(\tau _{n}v_{n}) =\frac{1}{\mu }I_{\lambda }(\tau _{n}{v}_{n})-\frac{1}{p\vartheta \mu }\left\langle I_{\lambda }^{\prime }(\tau _{n}{v}_{n}),\tau _{n}{v}_{n}\right\rangle +o(1)\\&=\frac{1}{p\mu }\mathcal {M}\left( \int _{\mathbb {R}^{N}}\left|\tau _{n}\nabla {v}_{n}\right|^{p}\textrm{d}x\right) +\frac{1}{p\mu }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|\tau _{n}{v}_{n}\right|^{p}\,\textrm{d}x-\frac{\lambda }{\mu }\int _{\mathbb {R}^{N}}{H(x,\tau _{n}{v}_{n})}\,\textrm{d}x\\&\quad -\frac{1}{p\vartheta \mu }M\left( \int _{{\mathbb {R}}^N}\left|\tau _{n}\nabla {v}_{n}\right|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}\left|\tau _{n}\nabla {v}_{n}\right|^{p-2}\tau _{n}\nabla {v}_{n}\cdot \tau _{n}\nabla {v}_{n}\,\textrm{d}x\\&\quad -\frac{1}{p\vartheta \mu }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|\tau _{n}{v}_{n}\right|^{p}\,\textrm{d}x + \frac{\lambda }{p\vartheta \mu }\int _{\mathbb {R}^{N}}{h(x,\tau _{n}{v}_{n})\tau _{n}{v}_{n}}\,\textrm{d}x+o(1)\\&=\frac{1}{p\vartheta \mu }\left[ \vartheta \mathcal {M}\left( {\tau }^p_n\int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) -M\left( {\tau }^p_n\int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) \left( {\tau }^p_n\int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) \right] \\&\quad +{\tau }^{p}_{n}\left( \frac{1}{p\mu }-\frac{1}{p\vartheta \mu }\right) \int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|{v}_{n}\right|^{p}\textrm{d}x\\&\qquad \qquad +\frac{\lambda }{p\vartheta \mu }\int _{\mathbb {R}^{N}}\big (h(x,{\tau }_n{v}_{n})({\tau }_n{v}_{n})-p\vartheta H(x,{\tau }_n{v}_{n})\big )\,\textrm{d}x + o(1)\\&=\frac{1}{p\vartheta \mu }\widehat{\mathcal {M}}\left( {\tau }^p_n\int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) +{\tau }^{p}_{n}\left( \frac{1}{p\mu }-\frac{1}{p\vartheta \mu }\right) \int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|{v}_{n}\right|^{p}\textrm{d}x\\&\qquad \qquad +\frac{\lambda }{p\vartheta \mu }\int _{\mathbb {R}^{N}}\mathcal {H}(x,{\tau }_n{v}_{n})\,\textrm{d}x+o(1)\\&\le \frac{1}{p\vartheta }\widehat{\mathcal {M}}\left( \int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) +\left( \frac{1}{p}-\frac{1}{p\vartheta }\right) \int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|{v}_{n}\right|^{p}\textrm{d}x\\&\qquad \qquad +\frac{\lambda }{p\vartheta }\int _{\mathbb {R}^{N}}\mathcal {H}(x,{v}_{n})\,\textrm{d}x+K+o(1)\\&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) -\frac{1}{p\vartheta }M\left( \int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) \left( \int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) \\&\quad +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|{v}_{n}\right|^{p}\textrm{d}x-\frac{1}{p\vartheta }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|{v}_{n}\right|^{p}\textrm{d}x\\&\qquad \qquad +\lambda \int _{\mathbb {R}^{N}}\left( \frac{1}{p\vartheta }h(x,{v}_{n}){v}_{n}-H(x,{v}_{n})\right) \,\textrm{d}x+K+o(1)\\&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|{v}_{n}\right|^{p}\textrm{d}x-\lambda \int _{\mathbb {R}^{N}}H(x,{v}_{n})\,\textrm{d}x\\&\quad - \frac{1}{p\vartheta }\Big [M\left( \int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) \left( \int _{\mathbb {R}^{N}}\left|\nabla {v}_{n}\right|^{p}\textrm{d}x\right) +\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|{v}_{n}\right|^{p}\textrm{d}x\\&\qquad \qquad -\lambda \int _{\mathbb {R}^{N}}h(x,{v}_{n}){v}_{n}\,\textrm{d}x\Big ]+K+o(1)\\&=I_{\lambda }({v}_{n})-\frac{1}{p\vartheta }\left\langle I_{\lambda }^{\prime }({v}_{n}),{v}_{n}\right\rangle +K+o(1) \rightarrow \, c+K \quad \text {as} \quad n\rightarrow \infty , \end{aligned}$$

which is inconsistent with (2.9). Consequently we assert that the sequence $\{v_n\}$ is bounded in X. $\square$

Next we obtain the compactness condition as in Lemma 2.5 when the nonlinear term h holds the assumption ($\Psi 3$) instead of ($\Psi 2$).

Remark 2.6

In contrast to Lemma 2.5, the following compactness condition does not require the ($\Psi 6$) assumption, which is the behaviour for nonlinear terms h at infinity. The basic idea of proof for this result follows similar arguments in [14, 15]. From this point of view, it is important that the potential function $\mathfrak {P}\in C(\mathbb {R}^N, (0,\infty ))$ is coercive. As mentioned in the introduction, we prove this condition without assuming the coercivity of the function $\mathfrak {P}$.

Lemma 2.7

Assume that (K1)–(K2), (V), ($\Psi 1$) and ($\Psi 3$) hold. Then the functional $I_{\lambda }$ fulfils the $(C)_{c}$-condition for any $\lambda >0$.

Proof

For any $c \in \mathbb {R}$, let $\{v_{n}\}$ be a $(C)_c$-sequence in X satisfying (2.1). As in the proof of Lemma 2.5, it is enough to prove that $\{v_n\}$ is bounded in X. By means of the condition (V), we note that

$$\begin{aligned} \left( \frac{1}{\vartheta p}-\frac{1}{\mu }\right) \int _{\mathbb {R}^N}\mathfrak {P}(x)|v_n|^{p}\textrm{d}x&-C_{1}\int _{\{|v_n|\le T\}}\left( \sigma _{0}(x)|v_n|+b_{1}|v_n|^{q}\right) \textrm{d}x \nonumber \\&\ge \frac{1}{2}\left( \frac{1}{\vartheta p}-\frac{1}{\mu }\right) \int _{\mathbb {R}^N}\mathfrak {P}(x)|v_n|^{p}\textrm{d}x-\widetilde{\mathfrak {K}}_{0} \end{aligned}$$

(2.10)

for any positive constant $C_{1}$ and for some positive constant $\mathfrak {K}_{0}$. Indeed, without loss of generality, suppose that $T>1$. By Young’s inequality we know that

$$\begin{aligned}&\Big (\frac{1}{\vartheta p}-\frac{1}{\mu }\Big )\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x- C_{1} \int _{\{|v_n|\le T\}}\left( \sigma _{0}(x)|v_n|+b_{1}|v_n|^{q}\right) \,\textrm{d}x \nonumber \\&\ge \Big (\frac{1}{\vartheta p}-\frac{1}{\mu }\Big )\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x- C_{1} \int _{\{|v_{n}|\le T\}}\left( \sigma _{0}^{\gamma '}(x)+|v_n|^{\gamma }+b_{1}|v_n|^{q}\right) \,\textrm{d}x \nonumber \\&\ge \frac{1}{2}\Big (\frac{1}{\vartheta p}-\frac{1}{\mu }\Big )\left[ \int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x+\int _{\{|v_{n}|\le T\}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x\right] \nonumber \\&\quad -C_{1} \int _{\{|v_{n}|\le 1\}}\left( |v_n|^{p}+b_{1}|v_n|^{q}\right) \,\textrm{d}x\nonumber \\&\quad -C_{1} \int _{\{1<|v_{n}|\le T\}}\left( |v_n|^{\gamma }+b_{1}|v_n|^{q}\right) \,\textrm{d}x-C_{1}|\!|\sigma _{0}|\!|_{L^{\gamma ^{\prime }}(\mathbb {R}^N)}^{\gamma '} \nonumber \\&\ge \frac{1}{2}\Big (\frac{1}{\vartheta p}-\frac{1}{\mu }\Big )\left[ \int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x+\int _{\{|v_{n}|\le T\}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x\right] \nonumber \\&\quad -C_{1} \left( 1+b_{1}\right) \int _{\{|v_{n}|\le 1\}}\left|v_{n}\right|^{p}\,\textrm{d}x-C_{1}|\!|\sigma _{0}|\!|_{L^{p^{\prime }}(\mathbb {R}^N)}^{p'}\nonumber \\&\quad -C_{1} \left( T^{\gamma -p}+T^{q-p}b_{1}\right) \int _{\{1<|v_{n}|\le T\}}\left|v_{n}\right|^{p}\,\textrm{d}x \nonumber \\&\ge \frac{1}{2}\Big (\frac{1}{\vartheta p}-\frac{1}{\mu }\Big )\left[ \int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x+\int _{\{|v_{n}|\le T\}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x\right] \nonumber \\&\quad -C_{2} \int _{\{|v_{n}|\le T\}}|v_{n}|^{p}\,\textrm{d}x-\widetilde{C}_1, \end{aligned}$$

(2.11)

where $C_{2}:=C_{1} \left( T^{\gamma -p}+T^{q-p}b_{1}\right)$ and $\widetilde{C}_1:=C_{1}|\!|\sigma _{0}|\!|_{L^{\gamma ^{\prime }}(\mathbb {R}^N)}^{\gamma '}$. Set

$$\begin{aligned} \mathbb {B}_{r_0}=\{x \in \mathbb {R}^{N}: |x| < r_0\},\quad A=\{x\in \mathbb {R}^{N}\setminus \mathbb {B}_{r_{0}}: \mathfrak {P}(x)\ge P_{0}\} \end{aligned}$$

and

$$\begin{aligned} B=\{x\in \mathbb {R}^{N}\setminus \mathbb {B}_{r_{0}}: \mathfrak {P}(x)<P_{0}\} \end{aligned}$$

for any $P_{0}>0$. Then it is clear that $A\cup B=\mathbb {B}_{r_{0}}^{c}$ where the sets A and B are disjoint. If $x\in A$, then for any $P_{0}\ge \frac{2\vartheta p\mu C_2}{\mu -\vartheta p}$, we know that

$$\begin{aligned} \mathfrak {P}(x)|v_{n}|^{p}\ge \frac{2\vartheta p\mu C_2}{\mu - \vartheta p}|v_{n}|^{p} \end{aligned}$$

(2.12)

for $|x|\ge r_0$. Also, since $\mathfrak {P} \in L^1_{loc}(\mathbb {R}^{N})$, we infer

$$\begin{aligned} \int _{\{|v_{n}|\le T\}\cap \mathbb {B}_{r_0}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x\le C_{3} \quad \text {and}\quad \int _{\{|v_{n}|\le T\}\cap \mathbb {B}_{r_0}}|v_{n}|^{p}\,\textrm{d}x\le C_{4} \end{aligned}$$

(2.13)

for some positive constants $C_{3}$, $C_{4}$. Using (V), we have $\mathfrak {C}_0=\text {meas}(\{|v_{n}|\le T\}\cap B)$ is finite and thus

$$\begin{aligned} 0\le \int _{\{|v_{n}|\le T\}\cap B}\mathfrak {P}(x)|v_{n}|^{p}\textrm{d}x\quad \text {and}\quad \int _{\{|v_{n}|\le T\}\cap B}|v_{n}|^{p}\textrm{d}x\le \mathfrak {C}_{0}T^{p}. \end{aligned}$$

(2.14)

This together with (2.11)–(2.14) yields

$$\begin{aligned}&\Big (\frac{1}{\vartheta p}-\frac{1}{\mu }\Big )\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x- C_{1} \int _{\{|v_{n}|\le T\}}(\sigma _{0}(x)|v_n|+b_{1}|v_n|^{q})\,\textrm{d}x \\&\ge \frac{\mu -\vartheta p}{2\vartheta p\mu }\Big [\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x+\int _{\{|v_{n}|\le T\}\cap \mathbb {B}^c_{r_0}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x+ \int _{\{|v_{n}|\le T\}\cap \mathbb {B}_{r_0}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x\Big ]\\&\quad -C_{2} \Big [\int _{\{|v_{n}|\le T\}\cap \mathbb {B}^c_{r_0}}|v_{n}|^{p}\,\textrm{d}x + \int _{\{|v_{n}|\le T\}\cap \mathbb {B}_{r_0}}|v_{n}|^{p}\,\textrm{d}x\Big ]-\widetilde{C}_1\\&\ge \frac{\mu -\vartheta p}{2\vartheta p\mu }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x+\frac{\mu -\vartheta p}{2\vartheta p\mu }\int _{\{|v_{n}|\le T\}\cap A}\mathfrak {P}(x)|v_{n}|^{p}\textrm{d}x \\&\quad -C_{2}\int _{\{|v_{n}|\le T\}\cap A}|v_{n}|^{p}\textrm{d}x+\frac{\mu -\vartheta p}{2\vartheta p\mu }\int _{\{|v_{n}|\le T\}\cap B}\mathfrak {P}(x)|v_{n}|^{p}\textrm{d}x\\&\quad -C_{2}\int _{\{|v_{n}|\le T\}\cap B}|v_{n}|^{p}\textrm{d}x-\mathfrak {K}_{0}\\&\ge \frac{\mu -\vartheta p}{2\vartheta p\mu }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x+\frac{\mu -\vartheta p}{2\vartheta p\mu }\int _{\{|v_{n}|\le T\}\cap B}\mathfrak {P}(x)|v_{n}|^{p}\textrm{d}x\\&\quad -C_{2}\int _{\{|v_{n}|\le T\}\cap B}|v_{n}|^{p}\textrm{d}x-\mathfrak {K}_0\\&\ge \frac{\mu -\vartheta p}{2\vartheta p\mu }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x-C_{2}\int _{\{|v_{n}|\le T\}\cap B}|v_{n}|^{p}\textrm{d}x-\mathfrak {K}_0\\&\ge \frac{1}{2}\Big (\frac{1}{\vartheta p}-\frac{1}{\mu }\Big )\int _{\mathbb {R}^{N}}\mathfrak {P}(x)|v_{n}|^{p}\,\textrm{d}x-\widetilde{\mathfrak {K}}_0, \end{aligned}$$

as claimed. From (K1)–(K2), ($\Psi 1$), ($\Psi 3$) and (2.10), one has

$$\begin{aligned} c+1&\ge I_{\lambda }(v_{n})-\frac{1}{\mu }\left\langle I_{\lambda }^{\prime }(v_{n}),v_{n}\right\rangle \\&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x\right) + \frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x \\&\quad -\lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x-\frac{1}{\mu }M\left( \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x \\&\quad -\frac{1}{\mu }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x+ \frac{\lambda }{\mu }\int _{\mathbb {R}^{N}}{h(x,v_{n})v_{n}}\,\textrm{d}x \\&\ge \frac{1}{p\vartheta }M\left( \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x + \frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x \\&\quad -\lambda \int _{\mathbb {R}^{N}}{H(x,v_{n})}\,\textrm{d}x-\frac{1}{\mu }M\left( \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x\\&\quad -\frac{1}{\mu }\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x+ \frac{\lambda }{\mu }\int _{\mathbb {R}^{N}}{h(x,v_{n})v_{n}}\,\textrm{d}x \\&\ge \left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) M\left( \int _{\mathbb {R}^{N}}|\nabla v_{n}|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\,\textrm{d}x\\&\quad +\left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) \int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x\\&\quad +\frac{\lambda }{\mu }\int _{\mathbb {R}^N}\left( h(x,v_n)v_n-\mu H(x,v_n)\right) \,\textrm{d}x \\&\ge m_{0}\left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\,\textrm{d}x +\left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) \int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x\\&\quad +\frac{\lambda }{\mu }\int _{\{|v_n|\le T\}}\left( h(x,v_n)v_n-\mu H(x,v_n)\right) \,\textrm{d}x\\&\quad +\frac{\lambda }{\mu }\int _{\{|v_n|\ge T\}}\left( h(x,v_n)v_n-\mu H(x,v_n)\right) \,\textrm{d}x \\&\ge m_{0}\left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\,\textrm{d}x +\left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) \int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x\\&\quad -C_{1}\int _{\{|v_n|\le T\}}\left( \sigma _{0}(x)|v_n|+b_{1}|v_n|^{q}\right) \textrm{d}x\\&\ge m_{0}\left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) \int _{\mathbb {R}^{N}}\left|\nabla v_{n}\right|^{p}\,\textrm{d}x +\frac{1}{2}\left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) \int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v_{n}\right|^{p}\,\textrm{d}x -\widetilde{\mathfrak {K}}_{0} \\&\ge \min \left\{ m_{0},\frac{1}{2}\right\} \left( \frac{1}{p\vartheta }-\frac{1}{\mu }\right) |\!|v_n|\!|_{X}^{p}-\widetilde{\mathfrak {K}}_0. \end{aligned}$$

Hence we conclude that the sequence $\{v_n\}$ is bounded in X. $\square$

Let $\mathfrak {H}$ be a separable and reflexive Banach space. Then it is known (see [6, 30]) that there are $\{e_{n}\}\subseteq \mathfrak {H}$ and $\{h_{n}^{*}\}\subseteq \mathfrak {H}^{*}$ such that

$$\begin{aligned} \mathfrak {H}=\overline{\text {span}\{e_{n}:n=1,2,\cdots \}},\ \ \mathfrak {H}^{*}=\overline{\text {span}\{h_{n}^{*}:n=1,2,\cdots \}}, \end{aligned}$$

and

$$\begin{aligned} \left\langle h^{*}_{i},e_{j}\right\rangle =\left\{ \begin{array}{ll} 1 &{}if \ i=j \\ 0 &{}if \ i\ne j. \end{array} \right. \end{aligned}$$

Let us denote $\mathfrak {H}_{n}=\text {span}\{e_{n}\}$, $\mathfrak {Y}_{k}=\bigoplus _{n=1}^{k}\mathfrak {H}_{n}$, and $\mathfrak {Z}_{k}= \overline{\bigoplus _{n=k}^{\infty }\mathfrak {H}_{n}}$.

Definition 2.8

Suppose that $(E,|\!|\cdot |\!|)$ is a real reflexive and separable Banach space, $\mathcal {H}\in C^{1}(E,\mathbb {R})$, $c\in \mathbb {R}$. We say that $\mathcal {H}$ fulfils the $(C)_c^*$-condition (with respect to $\mathfrak {Y}_n$) if any sequence $\{v_{n}\}_{n\in \mathbb {N}}\subset E$ for which $v_n\in \mathfrak {Y}_n$, for any $n\in \mathbb {N}$,

$$\begin{aligned} \mathcal {H}(v_n) \rightarrow c \quad \ \text{ and } \ \quad |\!|(\mathcal {H}|_{\mathfrak {Y}_n})^{\prime }(v_{n})|\!|_{E^{*}}(1+|\!|v_{n}|\!|) \rightarrow 0 \ \text {as} \ n\rightarrow \infty , \end{aligned}$$

has a subsequence converging to a critical point of $\mathcal {H}$.

Proposition 2.9

([12]) Suppose that $(E,|\!|\cdot |\!|)$ is a separable and reflexive Banach space, $\mathcal {H} \in C^{1}(E,\mathbb {R})$ is an even functional. If there is $k_0>0$ so that, for each $k\ge k_0$, there exist $\beta _{k}> \alpha _{k}>0$ such that

• (D1) $\inf \{\mathcal {H}(z):|\!|z|\!|=\beta _{k}, z\in \mathfrak {Z}_{k}\}\ge 0$;

• (D2) $\delta _{k}:=\max \{\mathcal {H}(z):|\!|z|\!|=\alpha _{k}, z\in \mathfrak {Y}_{k}\}<0$;

• (D3) $\phi _{k}:=\inf \{\mathcal {H}(z):|\!|z|\!|\le \beta _{k}, z\in \mathfrak {Z}_{k}\}\rightarrow 0$ as $k\rightarrow \infty$;

• (D4) $\mathcal {H}$ satisfies the $(C)_{c}^{*}$-condition for every $c\in [\phi _{k_{0}},0)$,

then $\mathcal {H}$ admits a sequence of negative critical values $c_{n}<0$ satisfying $c_{n}\rightarrow 0$ as $n\rightarrow \infty$.

Lemma 2.10

Let (K1)–(K2), (V), ($\Psi 1$)–($\Psi 2$) and ($\Psi 6$) hold. Then the functional $I_{{\lambda }}$ ensures the $(C)_{c}^{*}$-condition for any $\lambda >0$.

Proof

For $c \in \mathbb {R}$, let the sequence $\{v_{n}\}$ in X be such that $v_{n} \in \mathfrak {Y}_n$, for any $n\in \mathbb {N}$,

$$\begin{aligned} I_{\lambda }(v_n) \rightarrow c \quad \ \text{ and } \ \quad |\!|(I_{\lambda }|_{\mathfrak {Y}_n})^{\prime }(v_{n})|\!|_{X^{*}}(1+|\!|v_{n}|\!|_{X}) \rightarrow 0 \ \text {as} \ n\rightarrow \infty . \end{aligned}$$

Therefore, we get $c = I_{\lambda }(v_{n})+o_{n}(1)$ and $\left\langle I^{\prime }_{\lambda }(v_{n}),v_{n}\right\rangle =o_{n}(1),$ where $o_{n}(1) \rightarrow 0$ as $n \rightarrow \infty$. Repeating the argument from the proof of Lemma 2.5, we arrive that the sequence $\{v_{n}\}$ is bounded in X. So, there is a subsequence, still denoted by $\{v_{n}\}$, and a function $v_0$ in X such that $v_{n}\rightharpoonup v_{0}$ in X as $n\rightarrow \infty$.

To complete this proof, we will show that $v_n \rightarrow v_0$ in X as $n \rightarrow \infty$ and also $v_0$ is a critical point of $I_{\lambda }$. Even if the idea of this proof follows that in [12, Lemma 3.12], we give the proof for the sake of the convenience of the readers. As $X=\overline{\bigcup _{n \in \mathbb {N}}\mathfrak {Y}_{n}}$, we can choose $w_{n} \in \mathfrak {Y}_n, n \in \mathbb {N}$, such that $w_{n} \rightarrow v_0$ as $n \rightarrow \infty$. Since $|\!|(I_{\lambda }|_{\mathfrak {Y}_{n}})^{\prime }(v_{n})|\!| _{X^{*}} \rightarrow 0$, $\{v_{n}-w_{n}\}_{n\in \mathbb {N}}$ is bounded and $v_n-w_n \in \mathfrak {Y}_n$, we have

$$\begin{aligned} \langle I^{\prime }_{\lambda }(v_n),v_n-w_n\rangle =\langle (I_{\lambda }|_{\mathfrak {Y}_n})^{\prime }(v_{n}),v_n-w_n\rangle \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

(2.15)

Since $\Phi '$ is a continuous and bounded operator, it follows from Lemma 2.4 that $\{I^{\prime }_{\lambda }(v_n)\}$ is bounded because $\{v_n\}_{n\in \mathbb {N}}$ is bounded. Thus,

$$\begin{aligned} \langle I^{\prime }_{\lambda }(v_{n}),w_n-v_0\rangle \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

(2.16)

Using (2.15) and (2.16) we arrive that

$$\begin{aligned} \langle I^{\prime }_{\lambda }(v_{n}),v_n-v_0\rangle \rightarrow 0 \text { as } n \rightarrow \infty \end{aligned}$$

and so

$$\begin{aligned} \langle I^{\prime }_{\lambda }(v_{n})- I^{\prime }_{\lambda }(v_{0}),v_{n} - v_{0}\rangle \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

(2.17)

According to Lemma 2.4, we know

$$\begin{aligned} \langle \Psi ^{\prime }_{\lambda }(v_{n})-\Psi ^{\prime }_{\lambda }(v_{0}),v_n-v_0\rangle \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

(2.18)

From (2.17) and (2.18) we derive that

$$\begin{aligned} \langle \Phi ^{\prime }(v_{n})-\Phi ^{\prime }(v_{0}),v_n-v_0\rangle \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

Since $\Phi ^{\prime }$ is a mapping of $(S_+)$ by Lemma 2.3, we conclude that $v_n \rightarrow v_0$ as $n \rightarrow \infty$. Furthermore, we have $I^{\prime }_{\lambda }(v_n) \rightarrow I^{\prime }_{\lambda }(v_0)$ as $n \rightarrow \infty$. Let us prove that $v_0$ is a critical point of $I_{\lambda }$. Indeed, fix $n_0 \in \mathbb {N}$ and take any $z \in \mathfrak {Y}_{n_0}$. We have for $n \ge n_0$

$$\begin{aligned}{} & {} \langle I^{\prime }_{\lambda }(v_0),z\rangle =\langle I^{\prime }_{\lambda }(v_0)-I^{\prime }_{\lambda }(v_n),z\rangle +\langle I^{\prime }_{\lambda }(v_n),z\rangle \\{} & {} \quad =\langle I^{\prime }_{\lambda }(v_0)- I^{\prime }_{\lambda }(v_n),z\rangle +\langle (I_{\lambda }|_{\mathfrak {Y}_n})^{\prime }(v_{n}),z\rangle , \end{aligned}$$

so, passing the limit on the right side of the equation above, as $n \rightarrow \infty$, we obtain

$$\begin{aligned} \langle I^{\prime }_{\lambda }(v_0),z\rangle =0 \text { for all } z\in \mathfrak {Y}_{n_0}. \end{aligned}$$

As $n_0$ is taken arbitrarily and $\bigcup _{n \in \mathbb {N}}\mathfrak {Y}_{n}$ is dense in X, we have $I^{\prime }_{\lambda }(v_0)=0$ as required. Then, we conclude that $I_{\lambda }$ satisfies the $(C)_c^*$-condition for any $c\in \mathbb {R}$ and for any $\lambda >0$. $\square$

Lemma 2.11

Let (K1)–(K2), (V), ($\Psi 1$) and ($\Psi 3$) hold. Then the functional $I_{{\lambda }}$ holds the $(C)_{c}^{*}$-condition for any $\lambda >0$.

Proof

The proof is quite similar to that of Lemma 2.10. $\square$

The following consequences are crucial to prove our main result. The idea of their proofs basically follows from the recent works [14, 15]. For the sake of convenience of the readers, we give the proofs.

Lemma 2.12

Assume that (K1)–(K2), (V) and ($\Psi 1$) are satisfied. Then there is $k_0>0$, so that, for each $k\ge k_0$, there exists $\beta _{k}>0$ such that

$$\begin{aligned} \inf \{I_{{\lambda }}(v): v\in \mathfrak {Z}_k, |\!|v|\!|_{X}=\beta _k\}\ge 0. \end{aligned}$$

Proof

Let us denote

$$\begin{aligned} \nu _{1,k}=\sup _{|\!|y|\!|_{X}=1,y\in \mathfrak {Z}_k}|\!|y|\!|_{L^{\gamma }({\mathbb {R}}^N)},\quad \nu _{2,k}=\sup _{|\!|y|\!|_{X}=1,y\in \mathfrak {Z}_k}|\!|y|\!|_{L^{q}({\mathbb {R}}^N)}. \end{aligned}$$

Then, it is immediate to ensure that $\nu _{1,k}\rightarrow 0$ and $\nu _{2,k}\rightarrow 0$ as $k\rightarrow \infty$ (see [12]). Denote $\nu _{k}=\max \{\nu _{1,k},\nu _{2,k}\}$. From (K1), (K2), ($\Psi 1$) and the definition of $\nu _{k}$, it follows that

$$\begin{aligned} I_{{\lambda }}(v)&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^{N}}\left|\nabla v\right|^{p}\,\textrm{d}x\right) +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v\right|^{p}\,\textrm{d}x-\lambda \int _{\mathbb {R}^{N}}{H(x,v)}\,\textrm{d}x\\&\ge \frac{1}{p\vartheta }M\left( \int _{\mathbb {R}^{N}}\left|\nabla v\right|^{p}\,\textrm{d}x\right) \int _{\mathbb {R}^{N}}\left|\nabla v\right|^{p}\,\textrm{d}x\\&\qquad \qquad +\frac{1}{p}\int _{\mathbb {R}^{N}}\mathfrak {P}(x)\left|v\right|^{p}\,\textrm{d}x-\lambda \int _{\mathbb {R}^{N}}{H(x,v)}\,\textrm{d}x\\&\ge \frac{m_0}{p\vartheta }\int _{\mathbb {R}^{N}}\left|\nabla v\right|^{p}\textrm{d}x + \frac{1}{p}\int _{\mathbb {R}^N}\mathfrak {P}(x)|v|^{p}\textrm{d}x -\lambda \int _{\mathbb {R}^{N}}H(x,v)\textrm{d}x\\&\ge \frac{\min \left\{ m_0,\vartheta \right\} }{\vartheta p}|\!|v|\!|_{X}^{p}-\lambda \int _{\mathbb {R}^{N}}H(x,v)\,\textrm{d}x\\&\ge \frac{\min \left\{ m_0,\vartheta \right\} }{\vartheta p}|\!|v|\!|_{X}^{p}-\lambda |\!|\sigma _{0}|\!|_{L^{\gamma ^{\prime }}(\mathbb {R}^{N})}|\!|v|\!|_{L^{\gamma }(\mathbb {R}^{N})}-\frac{b_{1}\lambda }{q}|\!|v|\!|_{L^{q}({\mathbb {R}}^N)}^{q}\\&\ge \frac{\min \left\{ m_0,\vartheta \right\} }{\vartheta p}|\!|v|\!|_{X}^{p} -\lambda |\!|\sigma _{0}|\!|_{L^{\gamma '}(\mathbb {R}^N)}\nu _{k}|\!|v|\!|_{X}-\frac{b_{1}\lambda }{q}\nu _{k}^{q}|\!|v|\!|_{X}^{q}\\&\ge \frac{\min \left\{ m_0,\vartheta \right\} }{\vartheta p}|\!|v|\!|_{X}^{p} -\lambda |\!|\sigma _{0}|\!|_{L^{\gamma '}(\mathbb {R}^N)}\nu _{k}|\!|v|\!|_{X}-\frac{b_{1}\lambda }{q}\nu _{k}^{q}|\!|v|\!|_{X}^{2q} \end{aligned}$$

for large enough k and $|\!|v|\!|_{X}\ge 1$. Choose

$$\begin{aligned} \beta _{k}=\left[ \frac{2{b}_{1}\lambda \vartheta p\nu _{k}^{q}}{q \min {\{m_{0}, \vartheta \}}}\right] ^{\frac{1}{p-2q}}. \end{aligned}$$

Let $v\in \mathfrak {Z}_{k}$ with $|\!|v|\!|_{X}=\beta _{k}>1$ for k large enough. Then, we choose a $k_{0}\in {\mathbb {N}}$ such that

$$\begin{aligned} I_{{\lambda }}(v)&\ge \frac{\min \{m_0,\vartheta \}}{\vartheta p}|\!|v|\!|_X^p-\frac{b_1{\lambda }}{q}\nu _k^q|\!|v|\!|_X^{2q}-{\lambda }|\!|\sigma _0|\!|_{L^{\gamma '}(\mathbb {R}^{N})}\nu _k|\!|v|\!|_X\\&\ge \left( \frac{\min \{m_0,\vartheta \}}{\vartheta p}-\frac{b_1{\lambda }}{q}\nu _k^q|\!|v|\!|_X^{2q-p}\right) |\!|v|\!|_X^p-{\lambda }|\!|\sigma _0|\!|_{L^{\gamma '}(\mathbb {R}^{N})}\nu _k|\!|v|\!|_X\\&\ge \left( \frac{\min \{m_0,\vartheta \}}{\vartheta p}-\frac{b_1{\lambda }}{q}\nu _k^q\beta _{k}^{2q-p}\right) \beta _{k}^p-{\lambda }|\!|\sigma _0|\!|_{L^{\gamma '}(\mathbb {R}^{N})}\nu _k\beta _{k}\\&\ge \frac{\min \{m_0,\vartheta \}}{2\vartheta p}\beta _k^p-{\lambda }|\!|\sigma _0|\!|_{L^{\gamma '}(\mathbb {R}^{N})}\left[ \frac{2b_1{\lambda }\vartheta p}{q\min \{m_0,\vartheta \}}\right] ^{\frac{1}{p-2q}}\nu _k^{\frac{p-q}{p-2q}}\\&\ge 0 \end{aligned}$$

for all $k\in {\mathbb {N}}$ with $k\ge k_{0}$, since $\lim _{k\rightarrow \infty }\beta _{k}=\infty .$ Therefore,

$$\begin{aligned} \inf \{I_{{\lambda }}(v):v\in \mathfrak {Z}_{k}, |\!|v|\!|_{X}=\beta _{k}\}\ge 0. \end{aligned}$$

$\square$

Lemma 2.13

Assume that (K1)–(K2), (V), ($\Psi 1$) and ($\Psi 4$) hold. Then for each sufficiently large $k \in \mathbb {N}$, there exists $\alpha _{k}>0$ with $0<\alpha _k<\beta _k$ such that

• (1) $\delta _{k}:=\max \{ I_{{\lambda }}(v):v\in \mathfrak {Y}_{k}, |\!|v|\!|_{X}=\alpha _{k}\}< 0$;

• (2) $\phi _{k}:=\inf \{ I_{{\lambda }}(v):v\in \mathfrak {Z}_{k}, |\!|v|\!|_{X}\le \beta _{k}\}\rightarrow 0$ as $k\rightarrow \infty$,

where $\beta _{k}$ is given in Lemma 2.12.

Proof

(1): Because $\mathfrak {Y}_k$ is finite dimensional, all the norms are equivalent. Then we choose constants $\varsigma _{1,k}>0$ and $\varsigma _{2,k}>0$ such that

$$\begin{aligned} \varsigma _{1,k}|\!|v|\!|_{X}\le |\!|v|\!|_{L^{\kappa }(\eta ,{\mathbb {R}}^N)}\text { and } |\!|v|\!|_{L^{q}({\mathbb {R}}^N)}\le \varsigma _{2,k}|\!|v|\!|_{X} \end{aligned}$$

for any $v\in \mathfrak {Y}_{k}$. Let $v\in \mathfrak {Y}_{k}$ with $|\!|v|\!|_{X}\le 1$. From ($\Psi 1$) and ($\Psi 4$), there are $\mathfrak {C}_1, \mathfrak {C}_2 >0$ such that

$$\begin{aligned} H(x,\ell )\ge \mathfrak {C}_1\eta (x)|\ell |^{\kappa } -\mathfrak {C}_{2}|\ell |^{q} \end{aligned}$$

for almost all $x\in \mathbb {R}^N$ and for all $\ell \in \mathbb {R}$. Since

$$\begin{aligned} \int _{\mathbb {R}^{N}}\left|\nabla v\right|^{p}\,\textrm{d}x\le \mathfrak {C}_3 \end{aligned}$$

for a positive constant $\mathfrak {C}_3$, one has

$$\begin{aligned} I_{{\lambda }}(v)&= \frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^N}\left|\nabla v\right|^{p}\,\textrm{d}x\right) + \frac{1}{p}\int _{\mathbb {R}^N}\mathfrak {P}(x)|v|^{p}\textrm{d}x - \lambda \int _{\mathbb {R}^N}H(x,v)\,\textrm{d}x\\&\le \frac{1}{p}\left( \sup _{0\le \xi \le \mathfrak {C}_3}M(\xi )\right) \int _{\mathbb {R}^N}\left|\nabla v\right|^{p}\,\textrm{d}x+\frac{1}{p}\int _{\mathbb {R}^N}\mathfrak {P}(x)|v|^{p}\textrm{d}x\\ {}&\qquad \qquad -\lambda \mathfrak {C}_{1}\int _{\mathbb {R}^N}\eta (x)|v|^{\kappa }\textrm{d}x+\lambda \mathfrak {C}_{2}\int _{\mathbb {R}^N}|v|^{q}\,\textrm{d}x\\&\le C_6|\!|v|\!|_{X}^{p}-\lambda \mathfrak {C}_{1}|\!|v|\!|_{L^{\kappa }(\eta ,\mathbb {R}^{N})}^{\kappa }+\lambda \mathfrak {C}_{2}|\!|v|\!|_{L^{q}(\mathbb {R}^{N})}^{q}\\&\le C_6|\!|v|\!|_{X}^{p}-\lambda \mathfrak {C}_{1}\varsigma _{1,k}^{\kappa }|\!|v|\!|_{X}^{\kappa }+\lambda \mathfrak {C}_{2}\varsigma _{2,k}^{q}|\!|v|\!|_{X}^{q} \end{aligned}$$

for some positive constant $C_{6}$. Let $g(\ell )=C_{6}\ell ^{p}-\lambda \mathfrak {C}_{1}{\varsigma _{1,k}^{\kappa }}{\ell }^{\kappa }+\lambda \mathfrak {C}_{2}{\varsigma _{2,k}^{q}}\ell ^{q}$. Since $\kappa<p<q$, we infer $g(\ell )<0$ for all $\ell \in (0,\ell _0)$ for sufficiently small $\ell _0\in (0,1)$. Hence we can find $\alpha _{k}>0$ such that $I_{{\lambda }}(v)<0$ for all $v\in \mathfrak {Y}_{k}$ with $|\!|v|\!|_{X}=\alpha _{k}< \ell _{0}$ for k large enough. If necessary, we can change $k_{0}$ to a large value, so that $\beta _{k}>\alpha _{k}>0$ and

$$\begin{aligned} \delta _{k}:=\max \{I_{{\lambda }}(v):v\in \mathfrak {Y}_{k},|\!|v|\!|_{X}=\alpha _{k}\}<0. \end{aligned}$$

(2): Since $\mathfrak {Y}_{k}\cap \mathfrak {Z}_{k}\ne \emptyset$ and $0<\alpha _{k}<\beta _{k}$, we get $\phi _{k}\le \delta _{k}<0$ for all $k\ge k_{0}$. For any $v\in \mathfrak {Z}_{k}$ with $|\!|v|\!|_{X}=1$ and $0<\tau <\beta _{k}$, one has

$$\begin{aligned} I_{{\lambda }}(\tau v)&=\frac{1}{p}\mathcal {M}\left( \int _{\mathbb {R}^N}\left|\nabla \tau v\right|^{p}\,\textrm{d}x\right) +\frac{1}{p}\int _{\mathbb {R}^N}\mathfrak {P}(x)|\tau v|^{p}\textrm{d}x -\lambda \int _{\mathbb {R}^N}H(x,\tau v)\,\textrm{d}x\\&\ge -\lambda |\!|\sigma _{0}|\!|_{L^{\gamma '}(\mathbb {R}^{N})}|\!|\tau v|\!|_{L^{\gamma }(\mathbb {R}^{N})} -\frac{b_{1}\lambda }{q}|\!|\tau v|\!|_{L^{q}({\mathbb {R}}^N)}^{q}\\&\ge -\lambda |\!|\sigma _{0}|\!|_{L^{\gamma '}(\mathbb {R}^{N})}\beta _{k}\nu _{k} -\frac{b_{1}\lambda }{q}\beta _{k}^{q} \nu _{k}^{q} \end{aligned}$$

for sufficiently large k, where $\nu _{k}$ was given in Lemma 2.12. Hence, from the definition of $\beta _{k}$ we infer

$$\begin{aligned} 0>\phi _{k}&\ge -\lambda |\!|\sigma _{0}|\!|_{L^{\gamma '}(\mathbb {R}^{N})}\beta _{k}\nu _{k}-\frac{b_{1}\lambda }{q}\beta _{k}^{q} \nu _{k}^{q}\\&= -\lambda |\!|\sigma _{0}|\!|_{L^{\gamma '}(\mathbb {R}^{N})}\left[ \frac{2{b}_{1}\lambda \vartheta p}{q\min {\{m_0,\vartheta \}}}\right] ^{\frac{1}{p-2q}} \nu _{k}^{\frac{p-q}{p-2q}}\\&\quad -\frac{b_{1}\lambda }{q}\left[ \frac{2{b}_{1}\lambda \vartheta p}{q\min {\{m_0,\vartheta \}}}\right] ^{\frac{q}{p-2q}} \nu _{k}^{\frac{q(p-q)}{p-2q}}. \end{aligned}$$

Because $p<q$, $q+p<2q$ and $\nu _{k}\rightarrow 0$ as $k\rightarrow \infty$, we assert that $\lim _{k\rightarrow \infty }\phi _k=0$. $\square$

We are ready to obtain our main results. With the aid of Lemmas 2.10–2.13, our final consequences are formulated as follows:

Theorem 2.14

Suppose that (K1)–(K2), (V), ($\Psi 1$)–($\Psi 2$), ($\Psi 4$) and ($\Psi 6$) are satisfied. If $h(x,-\ell )=-h(x,\ell )$ holds for all $(x,\ell )\in {\mathbb {R}}^N\times {\mathbb {R}}$, then for all ${\lambda }>0$ the problem $(P_{\lambda })$ admits a sequence of nontrivial solutions $\{v_n\}$ in X satisfying $I_{{\lambda }}(v_{n})\rightarrow 0$ as $n\rightarrow \infty$.

Proof

Due to Lemma 2.10, we note that the functional $I_{{\lambda }}$ is even and fulfils the $(C)_c^*$-condition for every $c\in [\phi _{k_0},0)$. Now from lemmas 2.12 and 2.13, we ensure that properties ($D_1$), ($D_2$) and ($D_3$) in the dual fountain theorem hold. Therefore, the problem $(P_{\lambda })$ admits a sequence of weak solutions $\left\{ v_{n}\right\}$ with large enough n. The proof is complete. $\square$

Theorem 2.15

Suppose that (K1)–(K2), (V), ($\Psi 1$), ($\Psi 3$) and ($\Psi 4$) are satisfied. If $h(x,\ell )$ is odd in $\ell \in {\mathbb {R}}$, then for all ${\lambda }>0$ the problem $(P_{\lambda })$ admits a sequence of nontrivial solutions $\{v_n\}$ in X satisfying $I_{{\lambda }}(v_{n})\rightarrow 0$ as $n\rightarrow \infty$.

Proof

By a similar fashion, instead of Lemma 2.10, by Lemma 2.11, we can obtain this conclusion. $\square$