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On multiple solutions to a nonlocal fractional \(p(\cdot )\)-Laplacian problem with concave–convex nonlinearities

Abstract

The aim of this paper is to examine the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional \(p(\cdot )\)-Laplacian with concave–convex nonlinearities when, in general, the nonlinear term does not satisfy the Ambrosetti–Rabinowitz condition. The main tools for obtaining this result are the mountain pass theorem and a modified version of Ekeland’s variational principle for an energy functional with the compactness condition of the Palais–Smale type, namely the Cerami condition. Also we discuss several existence results of a sequence of infinitely many solutions to our problem. To achieve these results, we employ the fountain theorem and the dual fountain theorem as main tools.

Introduction

In the last years the study of problems involving differential equations and variational problems associated with the \(p(\cdot )\)-Laplacian operator has been paid to an increasing deal of attention because they can be viewed as a model for many physical phenomena which arise in several investigations related to elastic mechanics, electro-rheological fluid (“smart fluids”), image processing, etc. We refer the reader to [6, 16, 21, 32, 43, 49] and the references therein.

On the other hand, in the recent years the study of equations with nonstandard growth and related nonlocal equations has gained an increasing deal of attention due to both pure mathematical research aspects and real-world applications. This fact is justified by the occurrence of the aforementioned problems in many different applications such as conservation laws, ultra-materials and water waves, phase transitions, thin obstacle problem, optimization, flames propagation, stratified materials, anomalous diffusion, ultra-relativistic limits of quantum mechanics, crystal dislocation, soft thin films, minimal surfaces, semipermeable membranes and flame propagation, multiple scattering, mathematical finance, and so on. For a comprehensive introduction to the study of nonlocal problems, we refer to the work of Di Nezza, Palatucci, and Valdinoci [20], see [13, 25, 34, 36, 50] and the references therein for more details.

Therefore, a natural question is to understand if some results can be recovered when we change the local \(p(\cdot )\)-Laplacian, defined as \(-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)\), into the nonlocal fractional \(p(\cdot )\)-Laplacian. In this direction, several researchers have attempted to extend the study of the classical exponent variable case to include the fractional case (see for instance [4, 7, 8, 26, 27, 31, 38, 58]). In particular, as far as we are aware, Kaumann et al. [31] defined a new class of fractional Sobolev spaces with variable exponents that takes a fractional variable exponent operator into consideration. In particular, in [8] the authors discussed several fundamental properties related to the aforementioned function space and the related nonlocal operator and, using a direct variational method, the authors showed an application to a class of nonlocal fractional problems with several variable exponents. Precisely, as applications, they proved the existence of at least one solution for equations driven by the fractional \(p(\cdot )\)-Laplacian. Inspired by these recent works, further fundamental embeddings for the fractional Sobolev spaces with variable exponents and their applications—such as a priori bounds and multiplicity of solutions of problems driven by the fractional \(p(\cdot )\)-Laplacian—have been provided by Ho and Kim [26]. Also they obtained the existence of many solutions for a class of critical nonlocal problems with variable exponents; see [27]. We refer the interested reader to [4, 5, 58] for the existence results to Kirchhoff-type problems driven by a \(p(\cdot )\)-fractional operator.

This paper is devoted to the study of the existence of nontrivial solutions for the following Schrödinger-type problem involving the nonlocal fractional \(p(\cdot )\)-Laplacian:

$$ -\mathfrak{L}_{\mathcal{K}}z+V(x){ \vert z \vert }^{p(x,x)-2}z= \lambda a(x) \vert z \vert ^{r(x)-2}z+ f(x,z) \quad \text{in } \mathbb{R}^{N}, $$
(P)

where \(N \ge 2\), \(\lambda >0\) is a parameter, \(p: \mathbb{R}^{N}\times \mathbb{R}^{N} \to (1,\infty )\) is a continuous function satisfying \(p(x):=p(x,x)\) for all \(x\in \mathbb{R}^{N}\), \(r: \mathbb{R}^{N} \to (1,\infty )\) is continuous, V and a are suitable potential functions in \((0,\infty )\), and \(f:\mathbb{R}^{N}\times \mathbb{R}\to \mathbb{R}\) satisfies a Carathéodory condition. Here, \(\mathfrak{L}_{\mathcal{K}}\) stands for the following pointwise-defined nonlocal operator:

$$ \mathfrak{L}_{\mathcal{K}}z(x)=2 \int _{\mathbb{R}^{N}} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)-2}\bigl(z(x)-z(y)\bigr) \mathcal{K}(x,y)\,dy \quad \text{for all } x\in \mathbb{R}^{N}, $$

where \({p}\in C(\mathbb{R}^{N}\times \mathbb{R}^{N})\) is uniformly continuous such that p is symmetric, i.e., \({p}(x,y)={p}(y,x)\) for all \(x,y\in \mathbb{R}^{N}\); \(0< s<1\); \(1<\inf_{(x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N}}{p}(x,y)\leq \sup_{(x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N}}{p}(x,y)< \frac{N}{s}\); and \(\mathcal{K}:\mathbb{R}^{N}\times \mathbb{R}^{N}\to (0,+\infty )\) is a kernel function such that the following conditions are fulfilled:

(\(\mathcal{K}\)1):

\(m\mathcal{K}\in L^{1}(\mathbb{R}^{N}\times \mathbb{R}^{N})\), where \(m(x,y)=\min \{|x-y|^{p(x,y)},1\}\);

(\(\mathcal{K}\)2):

There exists a constant \(\theta _{0}>0\) such that \(\mathcal{K}(x,y)|x-y|^{N+sp(x,y)}\geq \theta _{0}\) for almost all \((x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N} \) and \(x\neq y\);

(\(\mathcal{K}\)3):

\(\mathcal{K}(x,y)=\mathcal{K}(y,x)\) for all \((x,y) \in \mathbb{R}^{N}\times \mathbb{R}^{N}\).

With the choice \(\mathcal{K}(x,y)=|x-y|^{-N-sp(x,y)}\), the operator \(\mathfrak{L}_{\mathcal{K}}\) becomes the fractional \(p(\cdot )\)-Laplacian operator \((-\Delta )^{s}_{p(\cdot)}\) defined as

$$ (- \Delta )_{p(x)}^{s} z(x) = \mathrm{P.V.} \int _{\mathbb{R}^{N}} \frac{ \vert z(x) - z(y) \vert ^{{p}(x,y)-2} (z(x) - z(y))}{ \vert x - y \vert ^{N+s{p}(x,y)}} \,dy, \quad x \in \mathbb{R}^{N}. $$

The first purpose of the present paper is to establish the existence of at least two distinct nontrivial solutions for Schrödinger-type problems involving the nonlocal fractional \(p(\cdot )\)-Laplacian in case where the nonlinear term is concave–convex. The primary tools for obtaining this result are the mountain pass theorem (see [3]) and a variant of Ekeland’s variational principle (see [6]) for an energy functional. We assume that this energy functional satisfies a Palais–Smale-type compactness condition, namely the Cerami condition. This kind of nonlinearity has been extensively studied since the seminal work of Ambrosetti, Brezis, and Cerami [2]. For elliptic equations with the concave–convex nonlinearity, we refer the reader also to [12, 14, 15, 19, 28, 5355] and the references therein. Precisely, the existence of multiple solutions for an elliptic problem of a nonhomogeneous fractional p-Kirchhoff-type, involving concave–convex nonlinearities, has been studied in [55]. By means of variational techniques and Ekeland’s variational principle, the authors in [28] obtained the existence of two nontrivial nonnegative solutions and infinitely many solutions for the following degenerated \(p(x)\)-Laplacian equations involving concave–convex type nonlinearities with two parameters:

$$ \textstyle\begin{cases} -\operatorname{div}(w(x) \vert \nabla z \vert ^{p(x)-2} \nabla z) =\lambda a(x) \vert z \vert ^{r(x)-2}z+ \mu b(x) \vert z \vert ^{q(x)-2}z &\text{in } \Omega , \\ z = 0 &\text{on } \partial \Omega , \end{cases} $$

where \(\Omega \subset \mathbb{R}^{N}\) is a bounded domain with a smooth boundary \(\partial \Omega , p, q, r \in C(\overline{\Omega }, (1,\infty ))\) with \(r(x)< p(x)< q(x)\) for all \(x\in \Omega \), w, a, b are measurable functions on Ω that are positives a.e. in Ω, and λ, μ are real parameters. Very recently, Biswas and Tiwari [11] investigated an elliptic problem involving nonlocal operator with variable exponents and concave–convex nonlinearity in a bounded domain with Dirichlet boundary condition. Biswas and Tiwari assumed the condition by Ambrosetti and Rabinowitz [3] (see [1, 24] for elliptic equations with variable exponents) and then employed the mountain pass theorem and Ekeland’s variational principle to obtain the multiplicity result.

As we known, the condition of Ambrosetti–Rabinowitz type in [3], that is, there exists a constant \(\theta >p\) such that

$$\begin{aligned}& 0< \theta F(x,\tau )\le f(x,\tau )\tau , \\& \quad \text{for all } \tau \in \mathbb{R}\setminus \{0\} \text{ and } x \in \mathbb{R}^{N}, \text{where } F(x,\tau )= \int _{0}^{\tau }f(x,t) \,dt, \end{aligned}$$
(1.1)

is essential in securing the boundedness of the Palais–Smale sequence of an energy functional. However, this condition is quite restrictive and removes several nonlinearities. For this reason, during the last few decades there have been extensive studies which tried to drop it; see [1, 16, 29, 30, 33, 34, 3840, 42, 44].

In that sense, our first aim is to discuss the existence of two nontrivial distinct solutions to problem (P) for the case of a combined effect of concave–convex nonlinearities when the nonlinear growth f does not satisfy the condition of Ambrosetti–Rabinowitz type. The main point in the present paper is to discuss the existence of multiple solutions to (P) under a new and mild assumption for the convex term f that does not satisfy (1.1) and is different from those studied in [1, 16, 29, 30, 33, 34, 3840, 42, 44]. In particular, we give some examples to demonstrate that this condition is not artificial. The main difficulty for obtaining the multiplicity result under this assumption on the convex term f is to verify the Cerami condition of the energy functional associated with (P). It is worth noting that we overcome it from the coercivity of the potential function V.

For recent developments in the context of concave–convex problems, we mention the work of Papageorgiou–Scapellato [45] where the authors studied nonlinear Robin problems driven by the p-Laplacian plus and indefinite potential in which the reaction exhibits the competing effects of a parametric concave (that is, \((p-1)\)-sublinear) term and of a convex (that is, \((p-1)\)-superlinear) term. In [45] the authors did not require the Ambrosetti–Rabinowitz condition and obtained a bifurcation-type theorem that describes the dependence of a set of positive solutions on the parameter \(\lambda >0\). In line with the contents of the paper [45], Papageorgiou and Scapellato [47] considered Robin problems driven by the \((p,q)\)-Laplacian plus an indefinite potential term and did not require the Ambrosetti–Rabinowitz condition for the reaction. We mention that in [45] there is no parameter and the authors, in addition to constant sign solutions, produced nodal solutions. Finally, we cite a variant of the classical concave–convex problem studied in [46]. Precisely, Papageorgiou and Scapellato in [46] studied a nonlinear resonant boundary value problem where there is no parameter, the convex term is replaced by a resonant (that is, \((p-1)\)-linear) term, and the concave contribution comes from the boundary condition.

The second main aim of this paper is to obtain several existence results of a sequence of infinitely many solutions to problem (P). First we are to discuss that multiple large energy solutions for problem (P) exist (see Theorem 3.12). The second is to establish that problem (P) possesses a sequence of infinitely many small energy solutions (see Theorem 3.16). The strategy of the proof for these consequences is based on the applications of variational tools such as the fountain theorem and the dual fountain theorem, which were initially built by the papers [9] and [10], respectively. Our study on such multiplicity results for nonlinear elliptic equations of variational type is particularly inspired by the contributions in recent works [18, 30, 38, 41, 48, 51] and the references therein. However, in some sense the proof of our consequence for multiple small energy solutions is different from that of the previous related works [10, 41, 51, 52]. To the best of our knowledge, while many authors are interested in the study of elliptic problems in both local and nonlocal cases, the present paper is the first endeavor to develop the existence results for the concave–convex-type problems driven by nonlocal fractional \(p(\cdot )\)-Laplacian.

This paper’s outline is the following: we firstly present some necessary preliminary knowledge of function spaces. Next we give the variational framework associated with problem (P), and then we establish the results about at least two distinct nontrivial solutions to the nonlocal fractional \(p(\cdot )\)-Laplacian with concave–convex nonlinearities by applying the mountain pass theorem and a variant of Ekeland’s variational principle for an energy functional with the Cerami condition. Finally, under suitable conditions on the convex term f, we carry out various existence results of infinitely many nontrivial solutions by employing the variational principle.

Preliminaries

In this section we present a natural functional framework associated with problem (P). We briefly recall some definitions and fundamental properties of the variable exponent Lebesgue spaces and a Lebesgue–Sobolev space of fractional type \(W^{s,q(\cdot ),p(\cdot ,\cdot )}(\Omega )\) which will be used throughout the paper. For further details on these spaces, we refer the reader to [4, 7, 8, 26, 27, 31, 58].

Set

$$ C_{+}(\overline{\Omega })= \Bigl\{ \ell \in C(\overline{\Omega }): \min_{x \in \overline{\Omega }}\ell (x)>1 \Bigr\} . $$

For any \(\ell \in C_{+}(\overline{\Omega })\), we define

$$ \ell _{+}=\sup_{x\in \Omega }\ell (x)\quad \text{and}\quad \ell _{-}=\inf_{x\in \Omega }\ell (x). $$

Let \(M(\Omega )\) be the vector space of all measurable functions from Ω into \(\mathbb{R}\). We identify two such functions which differ only on a Lebesgue-null set. Given \(h\in C_{+}(\overline{\Omega })\), the anisotropic Lebesgue space \(L^{h(\cdot )}(\Omega )\) is defined by

$$ L^{h(\cdot )}(\Omega )= \biggl\{ z\in M(\Omega ): \int _{\Omega } \vert z \vert ^{h(x)} \,dx< \infty \biggr\} .$$

We equip this space with the so-called Luxemburg norm defined by

$$ \Vert z \Vert _{L^{h(\cdot )}(\Omega )}=\inf \biggl[\vartheta >0: \int _{\Omega } \biggl\lvert \frac{z(x)}{\vartheta } \biggr\rvert ^{h(x)} \,dx \leq 1 \biggr].$$

In the anisotropic Lebesgue spaces the following Hölder inequality holds.

Lemma 2.1

([23, 35])

The space \(L^{h(\cdot )}(\mathbb{R}^{N})\) is a separable, uniformly convex Banach space, and its conjugate space is \(L^{h'(\cdot )}(\mathbb{R}^{N})\) where \(1/h(x)+1/h'(x)=1\). For any \(z\in L^{h(\cdot )}(\mathbb{R}^{N})\) and \(\omega \in L^{h'(\cdot )}(\mathbb{R}^{N})\), we have

$$ \biggl\vert \int _{\mathbb{R}^{N}} z\omega \,dx \biggr\vert \le \biggl( \frac{1}{h_{-}}+\frac{1}{(h')_{-}} \biggr) \Vert z \Vert _{L^{h(\cdot )}( \mathbb{R}^{N})} \Vert \omega \Vert _{L^{h'(\cdot )}(\mathbb{R}^{N})}\le 2 \Vert z \Vert _{L^{h(\cdot )}(\mathbb{R}^{N})} \Vert \omega \Vert _{L^{h'(\cdot )}( \mathbb{R}^{N})}. $$

Lemma 2.2

([23])

Let us consider the modular function

$$ \psi (z)= \int _{\mathbb{R}^{N}} \vert z \vert ^{h(x)} \,dx \quad \textit{for any } z \in L^{h(\cdot )}\bigl(\mathbb{R}^{N}\bigr). $$

Then we have

  1. (1)

    \(\psi (z)>1\) (=1; <1) if and only if \(\|z\|_{L^{h( \cdot )}(\mathbb{R}^{N})}>1\) (=1; <1), respectively;

  2. (2)

    If \(\|z\|_{L^{h(\cdot )}(\mathbb{R}^{N})}>1\), then \(\|z\|_{L^{h(\cdot )}(\mathbb{R}^{N})}^{h_{-}}\le \psi (z)\le \|z\|_{L^{h( \cdot )}(\mathbb{R}^{N})}^{h_{+}}\);

  3. (3)

    If \(\|z\|_{L^{h(\cdot )}(\mathbb{R}^{N})}<1\), then \(\|z\|_{L^{h(\cdot )}(\mathbb{R}^{N})}^{h_{+}}\le \psi (z)\le \|z\|_{L^{h( \cdot )}(\mathbb{R}^{N})}^{h_{-}}\).

Let Ω be a Lipschitz domain in \(\mathbb{R}^{N}\). Let \(s\in (0,1)\) and let \(p\in C(\overline{\Omega }\times \overline{\Omega }, (1,\infty ))\) be such that \(p(x,y) = p(y, x)\) for all \(x,y\in \overline{\Omega }\) and

$$ 1 < p^{-}:=\inf_{(x,y)\in \overline{\Omega }\times \overline{\Omega }}p(x,y) \le p^{+}:=\sup_{(x,y)\in \overline{\Omega }\times \overline{\Omega }}p(x,y) < +\infty . $$

For \(q \in C_{+}(\mathbb{R}^{N})\), define

$$ W^{s,q(\cdot ),p(\cdot ,\cdot )}\bigl(\mathbb{R}^{N}\bigr):= \biggl\{ z \in L^{q( \cdot )}\bigl(\mathbb{R}^{N}\bigr): \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{ \vert z(x)-z(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+sp(x,y)}} \,dx\,dy < + \infty \biggr\} , $$

and we set

$$ [z]_{s,p(\cdot ,\cdot )}\bigl(\mathbb{R}^{N}\bigr):=\inf \biggl\{ \lambda >0: \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{ \vert z(x)-z(y) \vert ^{p(x,y)}}{\lambda ^{p(x,y)} \vert x-y \vert ^{N+sp(x,y)}} \,dx\,dy < 1 \biggr\} . $$

Then \(W^{s,q(\cdot ),p(\cdot ,\cdot )}(\mathbb{R}^{N})\) endowed with the norm

$$ \Vert z \Vert _{s,q,p}:= \Vert z \Vert _{L^{q(\cdot )}(\mathbb{R}^{N})}+[z]_{s,p(\cdot , \cdot )} \bigl(\mathbb{R}^{N}\bigr) $$

is a separable reflexive Banach space (see [7, 8, 31]).

Lemma 2.3

([26])

Let Ω be a bounded Lipschitz domain, and let p, q, and s be as above. Assume furthermore that

$$ sp^{+}< N\quad \textit{and}\quad q(x)\ge p(x)\quad \textit{for all } x\in { \overline{\Omega }}. $$

Then the following embedding holds:

$$ W^{s,q(\cdot ),p(\cdot ,\cdot )}(\Omega ) \hookrightarrow \hookrightarrow L^{r(\cdot )}( \Omega )$$

for any \(r\in C_{+}(\overline{\Omega })\) such that \(r(x)< p^{*}_{s}(x):=\frac{Np(x,x)}{N-sp(x,x)}\) for all \(x\in \overline{\Omega }\).

For the sake of brevity, we write \(p(x)\) in place of \(p(x,x)\) for some cases, and hence \(p\in C_{+}(\mathbb{R}^{N})\). In addition, we write \(W^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})\) in place of \(W^{s,p(\cdot ),p(\cdot ,\cdot )}(\mathbb{R}^{N})\). We recall the following embeddings (see [26, Theorem 3.5]).

Lemma 2.4

Let \(s\in (0,1)\). Let \(p\in C_{+}(\mathbb{R}^{N}\times \mathbb{R}^{N})\) be a uniformly continuous and symmetric function with \(sp^{+}< N\). Then it holds that

  1. (i)

    \(W^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})\hookrightarrow L^{r(\cdot )}( \mathbb{R}^{N})\) for any uniform continuous function \(r\in C_{+}(\mathbb{R}^{N})\) fulfilling \(p(x,x)\leq r(x)\) for all \(x\in \mathbb{R}^{N}\) and \(\inf_{x\in \mathbb{R}^{N}}(p_{s}^{\ast }(x)-r(x))>0\);

  2. (ii)

    \(W^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})\hookrightarrow \hookrightarrow L_{\mathrm {loc}}^{r(\cdot )}(\mathbb{R}^{N})\) for any \(r\in C_{+}(\mathbb{R}^{N})\) with \(r(x)< p_{s}^{\ast }(x)\) for all \(x\in \mathbb{R}^{N}\).

In the following, let \(0< s<1\) and let \(p\in C_{+}(\mathbb{R}^{N}\times \mathbb{R}^{N})\) be a uniformly continuous and symmetric function such that \(sp^{+}< N\). Suppose that \(\mathcal{K}:\mathbb{R}^{N}\times \mathbb{R}^{N} \to (0,\infty )\) is a function with conditions (\(\mathcal{K}\)1)–(\(\mathcal{K}\)3). Let us denote with \(W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})\) the completion of \(C^{\infty }_{0}(\mathbb{R}^{N})\) with respect to the norm

$$ \Vert z \Vert _{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})}:= \Vert z \Vert _{L^{p( \cdot )}(\mathbb{R}^{N})}+ \vert z \vert _{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}( \mathbb{R}^{N})}, $$

where

$$ \vert z \vert _{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})} :=\inf \biggl\{ \lambda >0: \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{1}{\lambda ^{p(x,y)}} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy < 1 \biggr\} . $$

According to the basic idea in [23], we obtain the following result.

Lemma 2.5

Denote

$$ \chi (z)= \int _{\mathbb{R}^{N}} \vert z \vert ^{p(x)} \,dx+ \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)}\mathcal{K}(x,y) \,dx \,dy \quad \textit{for any } z\in W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}\bigl( \mathbb{R}^{N}\bigr). $$

Then we have

  1. (1)

    \(\chi (z)>1\) (=1; <1) if and only if \(\|z\|_{W_{ \mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})}>1\) (=1; <1), respectively;

  2. (2)

    If \(\|z\|_{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})}>1\), then \(\|z\|_{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})}^{p^{-}} \le \chi (z)\le \|z\|_{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}( \mathbb{R}^{N})}^{p^{+}}\);

  3. (3)

    If \(\|z\|_{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})}<1\), then \(\|z\|_{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})}^{p^{+}} \le \chi (z)\le \|z\|_{W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}( \mathbb{R}^{N})}^{p^{-}}\).

Throughout this paper, we denote \(\mathcal{X}:=W_{\mathcal{K}}^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})\), and let \(\mathcal{X}^{*}\) be a dual space of \(\mathcal{X}\). Furthermore, \(\langle \cdot ,\cdot \rangle \) denotes the pairing of \(\mathcal{X}\) and its dual \(\mathcal{X}^{*}\).

Main results

In this section, we show the multiplicity result of a weak solution to problem (P) by employing the variational principle.

Definition 3.1

We say that \(z\in \mathcal{X}\) is a weak solution of problem (P) if

$$\begin{aligned} &\int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)-2}\bigl(z(x)-z(y)\bigr) \bigl( \varphi (x)-\varphi (y)\bigr) \mathcal{K}(x,y) \,dx\,dy \\ &\qquad {}+ \int _{\mathbb{R}^{N}}V(x) \lvert z \rvert ^{p(x)-2}z\varphi \,dx \\ &\quad =\lambda \int _{\mathbb{R}^{N}}a(x) \lvert z \rvert ^{r(x)-2}z \varphi \,dx+ \int _{\mathbb{R}^{N}}f(x,z)\varphi \,dx \end{aligned}$$

for all \(\varphi \in \mathcal{X}\).

Let us define the functional \(\mathcal{A}: \mathcal{X} \to \mathbb{R}\) by

$$ \mathcal{A}(z)= \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy+ \int _{\mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert z \rvert ^{p(x)}} \,dx. $$

Then from [8, 31] it follows that \(\mathcal{A} \in C^{1}(\mathcal{X},\mathbb{R})\), and its Fréchet derivative is given by

$$\begin{aligned} \bigl\langle {\mathcal{A}^{\prime }(z),\varphi }\bigr\rangle =& \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)-2}\bigl(z(x)-z(y)\bigr) \bigl(\varphi (x)- \varphi (y)\bigr) \mathcal{K}(x,y) \,dx\,dy \\ &{}+ \int _{\mathbb{R}^{N}}V(x){ \lvert z \rvert ^{p(x)-2}z\varphi } \,dx. \end{aligned}$$

Let \(F(x,\tau )=\int _{0}^{\tau }f(x,s) \,ds\). Let us assume that

  1. (H)

    \(p, q, r\in C_{+}(\mathbb{R}^{N})\) and \(1< r_{-}\le r_{+}< p^{-}\le p^{+}< q_{-}\le q_{+}< p_{s}^{*}(x)\) for all \(x\in \mathbb{R}^{N}\).

  2. (V)

    \(V \in L_{\mathrm{loc}}^{1}(\mathbb{R}^{N})\), \(\operatorname*{ess\,inf}_{x\in \mathbb{R}^{N}} {V}(x)>0\), and \(\lim_{|x|\to \infty }V(x)=+\infty \).

  3. (A)

    \(0\le a\in L^{\frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) with \(\lvert \{x\in {\mathbb{R}}^{N} : a(x)\neq 0\} \rvert >0\), where \(|A|\) denotes the Lebesgue measure of a subset A of \(\mathbb{R}^{N}\).

  4. (F1)

    \(f: \mathbb{R}^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Carathéodory condition.

  5. (F2)

    There exists \(0\le b\in L^{1}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) such that

    $$ \bigl\lvert f(x,\tau ) \bigr\rvert \le b(x) \lvert \tau \rvert ^{q(x)-1} \quad \text{for almost all } (x,\tau )\in \mathbb{R}^{N}\times \mathbb{R}, $$

    where \(q \in C_{+}(\mathbb{R}^{N})\) and \(q(x) < p^{*}_{s} (x)\) for all \(x\in \mathbb{R}^{N}\).

  6. (F3)

    \(\lim_{ \lvert \tau \rvert \to \infty }{ \frac{F(x,\tau )}{ \lvert \tau \rvert ^{p^{+}}}}=\infty \) uniformly for almost all \(x\in \mathbb{R}^{N}\).

  7. (F4)

    There are \(\mu >p^{+}\), \(M>0\), and a function \(\varrho \ge 0\) with \(\varrho \in L^{\frac{p(\cdot )}{p(\cdot )-p^{-}}}(\mathcal{B}_{1})\) on \(\mathcal{B}_{1}:=\{x\in \mathbb{R}^{N}:p(x)>p^{-}\}\) and \(\varrho (x)\equiv \tilde{\varrho }\) (constant function) on \(\mathcal{B}_{2}:=\{x\in \mathbb{R}^{N}:p(x)= p^{-}\}\) such that \(\lvert \{x\in \mathbb{R}^{N}:\varrho (x)>0\} \rvert \ne 0\) and

    $$ \tau f(x,\tau )-\mu F(x,\tau )\geq -\varrho (x) \lvert \tau \rvert ^{p^{-}}-\zeta (x) $$

    for all \((x,\tau )\in \mathbb{R}^{N}\times \mathbb{R}\) with \(|\tau |\ge M\) and for some \(\zeta \in L^{1} (\mathbb{R}^{N} )\cap L^{\infty } ( \mathbb{R}^{N} )\) with \(\zeta (x)\ge 0\).

  8. (F5)

    \(F(x,\tau )=o(|\tau |^{p(x)})\) as \(\tau \to 0\) uniformly for all \(x\in \mathbb{R}^{N}\).

As mentioned in the introduction, assumption (F4) for the convex term f is different from that used in the works [1, 16, 29, 30, 33, 34, 3840, 42, 44]. Hence we give some simple examples of functions that satisfy condition (F4).

Example 3.2

If \(p(x)=2\) for all \(x\in \mathbb{R}^{N}\) and

$$ f(x,\tau )=\rho (x) \lvert \tau \rvert \bigl(4\tau ^{3}-2\tau \cos \tau -4\sin \tau \bigr),$$

where \(\rho (x)\in L^{1}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) and \(0<\inf_{x\in \mathbb{R}^{N}}\rho (x)\le \sup_{x\in \mathbb{R}^{N}} \rho (x)<\infty \), then

$$ F(x,\tau )=\rho (x) \biggl(\frac{4}{5} \lvert \tau \rvert ^{5}-2 \tau \lvert \tau \rvert \sin \tau \biggr).$$

We set \(\tilde{\varrho }:=\inf_{x\in \mathbb{R}^{N}}\rho (x)\) and \(\zeta (x):=2(\mu -2)\rho (x)\) with \(2<\mu <\frac{15}{4}\) for all \(x\in \mathbb{R}^{N}\). Then

$$\begin{aligned} f(x,\tau )\tau -\mu F(x,\tau )&=\rho (x) \biggl(4 \lvert \tau \rvert ^{5}-2 \lvert \tau \rvert ^{3}\cos \tau -4 \tau \lvert \tau \rvert \sin \tau -\frac{4}{5}\mu \lvert \tau \rvert ^{5}+2\mu \tau \lvert \tau \rvert \sin \tau \biggr) \\ &= \rho (x) \biggl(4 \lvert \tau \rvert ^{5} -\frac{4}{5} \mu \lvert \tau \rvert ^{5}-2 \lvert \tau \rvert ^{3} \cos \tau + (2\mu -4)\tau \lvert \tau \rvert \sin \tau \biggr) \\ &\ge \rho (x) \biggl(4 \lvert \tau \rvert ^{3} - \frac{4}{5}\mu \lvert \tau \rvert ^{3}-2 \lvert \tau \rvert \cos \tau - (2\mu -4) \biggr)\tau ^{2} \\ &\ge \rho (x) \biggl( \lvert \tau \rvert ^{3}+ \biggl(3- \frac{4}{5}\mu \biggr) \lvert \tau \rvert ^{3}-2 \lvert \tau \rvert - (2\mu -4) \biggr) \\ &\ge \rho (x) \lvert \tau \rvert ^{2}-(2\mu -4)\rho (x) \\ &\ge -\tilde{\varrho } \lvert \tau \rvert ^{2}-\zeta (x) \end{aligned}$$

for \(\lvert \tau \rvert \ge r\), where \(r>1\) is chosen such that \((3-\frac{4}{5}\mu )r^{3}-2r\ge 0\). Hence (F4) is fulfilled.

Example 3.3

If \(p(x)=p>1\) for all \(x\in \mathbb{R}^{N}\) and

$$ f(x,\tau )=\rho (x) \biggl( \lvert \tau \rvert ^{p-2}\tau + \frac{2}{p}\sin \tau \biggr), $$

where ρ comes from the previous example, then

$$ F(x,\tau )=\rho (x) \biggl( \frac{1}{p} \lvert \tau \rvert ^{p} - \frac{2}{p}\cos \tau +\frac{2}{p} \biggr). $$

We set \(\tilde{\varrho }:=(\mu -1)\sup_{x\in \mathbb{R}^{N}}\rho (x)\) and \(\zeta (x):=\frac{4\mu }{p}\rho (x)\) with \(p<\mu \) for all \(x\in \mathbb{R}^{N}\). Then

$$\begin{aligned} f(x,\tau )\tau -\mu F(x,\tau )&= \rho (x) \biggl[ \lvert \tau \rvert ^{p}+\frac{2}{p}\tau \sin \tau -\frac{\mu }{p} \lvert \tau \rvert ^{p} + \frac{2\mu }{p}\cos \tau - \frac{2\mu }{p} \biggr] \\ &\ge \rho (x) \biggl[ \biggl(1-\frac{\mu }{p} \biggr) \lvert \tau \rvert ^{p} - \frac{2}{p} \lvert \tau \rvert - \frac{4\mu }{p} \biggr] \\ &= \rho (x) \biggl[ (1-\mu ) \lvert \tau \rvert ^{p} + \frac{\mu (p-1)}{p} \lvert \tau \rvert ^{p} - \frac{2}{p} \lvert \tau \rvert \biggr] - \frac{4\mu }{p} \rho (x) \\ &\geq \rho (x) (1-\mu ) \lvert \tau \rvert ^{p} - \frac{4\mu }{p} \rho (x) \\ &\geq -\tilde{\varrho } \lvert \tau \rvert ^{p} - \zeta (x) \end{aligned}$$

for all \(\lvert \tau \rvert \geq r\), where \(r>1\) is chosen such that \(\mu (p-1)r^{p} - 2r \geq 0\). Hence (F4) is fulfilled.

Example 3.4

If \(p\in C_{+}(\mathbb{R}^{N})\) and

$$ f(x,\tau )=\rho (x) \lvert \tau \rvert ^{p(x)-1}\tau \bigl[\bigl(p(x)+3 \bigr)\tau ^{2}-2\bigl(p(x)+2\bigr) \lvert \tau \rvert +\bigl(p(x)+1 \bigr) \bigr],$$

where \(\rho (x)\in L^{1}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\), then

$$ F(x,\tau )=\rho (x) \bigl( \lvert \tau \rvert ^{p(x)+3}-2 \lvert \tau \rvert ^{p(x)+2}+ \lvert \tau \rvert ^{p(x)+1}\bigr).$$

We set \(\varrho (x):=\rho (x)=:\zeta (x)\) and \(p_{-}+1<\mu <p(x)+2\) for all \(x\in \mathbb{R}^{N}\). Then

$$\begin{aligned} &f(x,\tau )\tau -\mu F(x,\tau ) \\ &\quad =\rho (x) \bigl[\bigl(p(x)+3-\mu \bigr) \lvert \tau \rvert ^{p(x)+3}-2\bigl(p(x)+2-\mu \bigr) \lvert \tau \rvert ^{p(x)+2}+\bigl(p(x)+1-\mu \bigr) \lvert \tau \rvert ^{p(x)+1} \bigr] \\ &\quad \ge \rho (x) \bigl[\bigl(p(x)+3-\mu \bigr) \lvert \tau \rvert ^{2}-2\bigl(p(x)+2- \mu \bigr) \lvert \tau \rvert + \bigl(p^{-}+1-\mu \bigr) \bigr] \lvert \tau \rvert ^{p(x)+1} \\ &\quad = \rho (x) \bigl[ \lvert \tau \rvert ^{2}+\bigl(p(x)+2-\mu \bigr) \bigl( \lvert \tau \rvert ^{2}-2 \lvert \tau \rvert \bigr)+ \bigl(p^{-}+1-\mu \bigr) \bigr] \lvert \tau \rvert ^{p(x)+1} \\ &\quad \ge \rho (x) \bigl[ \lvert \tau \rvert ^{2}-\bigl(\mu -p^{-}-1\bigr) \bigr] \lvert \tau \rvert ^{p^{-}} \\ &\quad \ge -\varrho (x) \lvert \tau \rvert ^{p^{-}}-\zeta (x) \end{aligned}$$

for \(\lvert \tau \rvert \ge r\), where \(r>1+\sqrt{(\mu -p^{-}-1)}\) is chosen such that \(r^{2}-2r\ge 0\). Hence (F4) is fulfilled.

Let us define the functional \(\Psi _{{\lambda }}:\mathcal{X}\to \mathbb{R}\) by

$$ \Psi _{{\lambda }}(z)=\lambda \int _{\mathbb{R}^{N}}{\frac{a(x)}{r(x)} \lvert z \rvert ^{r(x)}} \,dx+ \int _{\mathbb{R}^{N}}F(x,z) \,dx. $$

It is easy to check that \(\Psi _{{\lambda }} \in C^{1}(\mathcal{X},\mathbb{R})\) and its Fréchet derivative is

$$ \bigl\langle \Psi _{{\lambda }}^{\prime }(z),\varphi \bigr\rangle =\lambda \int _{\mathbb{R}^{N}}a(x) \vert z \vert ^{r(x)-2}z\varphi \,dx+ \int _{ \mathbb{R}^{N}} f(x,z)\varphi \,dx $$

for any \(z,\varphi \in \mathcal{X}\). Next we define the functional \(\mathcal{I}_{\lambda }:\mathcal{X}\to \mathbb{R}\) by

$$ \mathcal{I}_{\lambda }(z)=\mathcal{A}(z)-\Psi _{{\lambda }}(z). $$

Then the functional \(\mathcal{I}_{\lambda }\in C^{1}(\mathcal{X},\mathbb{R})\) and its Fréchet derivative is

$$\begin{aligned} \bigl\langle \mathcal{I}_{\lambda }^{\prime }(z), \varphi \bigr\rangle ={}& \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)-2}\bigl(z(x)-z(y)\bigr) \bigl( \varphi (x)-\varphi (y)\bigr) \mathcal{K}(x,y) \,dx\,dy \\ &{}+ \int _{\mathbb{R}^{N}}V(x){ \lvert z \rvert ^{p(x)-2}z\varphi } \,dx -\lambda \int _{\mathbb{R}^{N}}a(x) \lvert z \rvert ^{r(x)-2}z \varphi \,dx- \int _{\mathbb{R}^{N}}f(x,z)\varphi \,dx \end{aligned}$$

for any \(z,\varphi \in \mathcal{X}\).

Under assumption (V), we can give the compact embedding.

Lemma 3.5

If the potential function V satisfies assumption (V), then

  1. (1)

    the embedding from \(\mathcal{X}\hookrightarrow L^{p(\cdot )}(\mathbb{R}^{N})\) is continuous and compact;

  2. (2)

    for any measurable function \(\ell :\mathbb{R}^{N}\to \mathbb{R}\) with \(p(x)<\ell (x)\) for all \(x\in \mathbb{R}^{N}\), there is a compact embedding \(\mathcal{X}\hookrightarrow L^{\ell (\cdot )}(\mathbb{R}^{N})\) if \(\inf_{x\in \mathbb{R}^{N}}(p_{s}^{*}(x)-\ell (x))>0\).

Proof

In order to prove this lemma, we can adapt the proof of Lemma 2.6 in [1]. For the case that the potential function V is coercive, we obtain a similar result involving variable exponents of fractional type using Lemma 2.4. So, we omit the details of the proof. □

Next we give the following useful lemmas which are essential in obtaining the existence of at least two distinct nontrivial solutions to problem (P).

Definition 3.6

Let E be a real Banach space with dual space \(E^{*}\), \(\mathcal{I}\in C^{1}(E,\mathbb{R}^{N})\). We say that \(\mathcal{I}\) satisfies the Cerami condition (\((C)\)-condition, for short) in E if any \((C)\)-sequence \(\{z_{n}\} \subset E\), i.e., \(\{\mathcal{I}(z_{n})\}\) is bounded and \(\|\mathcal{I}^{\prime }(z_{n})\|_{E^{*}}(1+\|z_{n}\|_{E})\to 0 \text{ as } n\to \infty \), has a convergent subsequence in E. We say that \(\mathcal{I}\) satisfies the Cerami condition at level c (\((C)_{c}\)-condition, for short) in E if any \((C)_{c}\)-sequence \(\{z_{n}\}\subset E\), i.e., \(\mathcal{I}(z_{n})\to c\) as \(n\to \infty \) and \(\|\mathcal{I}'(z_{n})\|_{E^{*}}(1+\|z_{n}\|_{E})\to 0\) as \(n\to \infty \), has a convergent subsequence in E.

Lemma 3.7

Assume that (H), (V), (A), and (F1)(F4) hold. Then the functional \(\mathcal{I}_{\lambda }\) satisfies the \((C)\)-condition for any \(\lambda >0\).

Proof

Let \(\{z_{n}\}\) be a \((C)\)-sequence in \(\mathcal{X}\) for \(\mathcal{I}_{\lambda }\), that is,

$$ \sup_{n \in \mathbb{N}} \bigl\lvert \mathcal{I}_{\lambda }(z_{n}) \bigr\rvert \leq \mathcal{M}_{0} \quad \text{and}\quad \bigl\langle \mathcal{I}_{\lambda }^{\prime }(z_{n}), z_{n} \bigr\rangle =o(1), $$
(3.1)

where \(o(1)\to 0\) as \(n\to \infty \), and \(\mathcal{M}_{0}\) is a positive constant. From Lemma 4.2 in [8] and Lemma 3.3 in [33], it follows that \(\mathcal{A}'\) and \(\Psi _{{\lambda }}'\) are of type \((S_{+})\). Since \(\mathcal{X}\) is a reflexive Banach space, it is enough to ensure that the sequence \(\{z_{n}\}\) is bounded in \(\mathcal{X}\). We argue by contradiction. Assume that the sequence \(\{ z_{n} \} \) is unbounded in \(\mathcal{X}\). So then we may suppose that \(\|z_{n}\|_{\mathcal{X}}>1\) and \(\|z_{n}\|_{\mathcal{X}}\to \infty \) as \(n\to \infty \). Let us denote the sequence \(\{ w_{n} \} \) with \(w_{n}={z_{n}}/{\|z_{n}\|_{\mathcal{X}}}\). Then, clearly, we have \(\{ w_{n} \} \subset \mathcal{X}\) and \(\|w_{n}\|_{\mathcal{X}}=1\). Hence, up to a subsequence, still denoted by \(\{ w_{n} \} \), we infer \(w_{n}\rightharpoonup \omega \) in \(\mathcal{X}\) as \(n\to \infty \) and by Lemma 3.5

$$ w_{n}(x) \to \omega (x)\quad \text{ a.e. in } \mathbb{R}^{N}\quad \text{and}\quad w_{n} \to \omega\quad \text{in }L^{\ell (\cdot )}\bigl(\mathbb{R}^{N}\bigr) \text{ as } n\to \infty $$
(3.2)

for \(p(x)\leq \ell (x)\) with \(\inf_{x\in \mathbb{R}^{N}}(p_{s}^{*}(x)-\ell (x))>0\). Notice that \(V(x)\to +\infty \) as \(|x|\to \infty \), then

$$\begin{aligned} & \int _{\mathbb{R}^{N}}V(x) \biggl(\frac{1}{p(x)}- \frac{1}{\mu } \biggr) \lvert z_{n} \rvert ^{p(x)} \,dx- C_{1} \int _{|z_{n}| \leq M}\bigl( \lvert z_{n} \rvert ^{p(x)}+b(x) \lvert z_{n} \rvert ^{q(x)}\bigr) \,dx \\ &\quad \geq \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}V(x) \lvert z_{n} \rvert ^{p(x)} \,dx- \mathcal{M}_{1}, \end{aligned}$$

where \(C_{1}\) and \(\mathcal{M}_{1} \) are positive constants. In fact we know that

$$\begin{aligned} & \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N}}V(x) \vert z_{n} \vert ^{p(x)}\,dx- C_{1} \int _{ \vert z_{n} \vert \le M} \vert z_{n} \vert ^{p(x)}+b(x) \vert z_{n} \vert ^{q(x)}\,dx \\ &\quad \ge \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}V(x) \vert z_{n} \vert ^{p(x)}\,dx +\frac{1}{2} \biggl( \frac{1}{p^{+}}- \frac{1}{\mu } \biggr) \int _{ \vert z_{n} \vert \le 1}V(x) \vert z_{n} \vert ^{p(x)}\,dx \\ &\qquad {} -C_{1} \int _{ \vert z_{n} \vert \le 1} \vert z_{n} \vert ^{p(x)}+b(x) \vert z_{n} \vert ^{q(x)}\,dx -C_{1} \int _{1< \vert z_{n} \vert \le M} \vert z_{n} \vert ^{p(x)}+b(x) \vert z_{n} \vert ^{q(x)}\,dx \\ &\quad \ge \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}V(x) \vert z_{n} \vert ^{p(x)}\,dx +\frac{1}{2} \biggl( \frac{1}{p^{+}}- \frac{1}{\mu } \biggr) \int _{ \vert z_{n} \vert \le 1}V(x) \vert z_{n} \vert ^{p(x)}\,dx \\ &\qquad {} -C_{1} \bigl(1+ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}\bigr) \int _{ \vert z_{n} \vert \le 1} \vert z_{n} \vert ^{p(x)}\,dx -\widetilde{C}_{1}, \end{aligned}$$

where \(C_{1}\) and \(\widetilde{C}_{1}\) are positive constants. Let us set \(\Sigma :=\{x\in \mathbb{R}^{N}:|z_{n}(x)|>1\}\). Since \(\lvert \Sigma \rvert \) is finite (\(|\cdot |\) is the Lebesgue measure in \(\mathbb{R}^{N}\)), \(\Sigma =\widetilde{\Sigma }\cup N\) where Σ̃ is a bounded set and N is of measure zero. Without loss of generality, suppose that there exists \(B_{r}(0)\subseteq \mathbb{R}^{N}\) such that \(\Sigma \subset B_{r}(0)\) where \(B_{r}(0)\) is the open ball centered at 0 with radius r in the Euclidean space \(\mathbb{R}^{N}\). Since \(V(x)\to +\infty \) as \(|x|\to \infty \), there is \(r_{0}>0\) such that \(|x|\ge r_{0}>r\) implies \(V(x)\ge 2C_{1}(1+\|\sigma \|_{L^{\infty }(\mathbb{R}^{N})}) \frac{\mu p^{+}}{\mu -p^{+}}\). Consequently, we get

$$\begin{aligned} & \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N}}V(x) \vert z_{n} \vert ^{p(x)}\,dx- C_{1} \int _{ \vert z_{n} \vert \le M} \vert z_{n} \vert ^{p(x)}+b(x) \vert z_{n} \vert ^{q(x)}\,dx \\ &\quad \ge \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}V(x) \vert z_{n} \vert ^{p(x)}\,dx +\frac{1}{2} \biggl( \frac{1}{p^{+}}- \frac{1}{\mu } \biggr) \int _{\Sigma ^{c}\cap B_{r_{0}}^{c}}V(x) \vert z_{n} \vert ^{p(x)}\,dx \\ &\qquad {} +\frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{ \Sigma ^{c}\cap B_{r_{0}}}V(x) \vert z_{n} \vert ^{p(x)}\,dx -C_{1} \bigl(1+ \Vert b \Vert _{L^{ \infty }(\mathbb{R}^{N})}\bigr) \int _{\Sigma ^{c}\cap B_{r_{0}}^{c}} \vert z_{n} \vert ^{p(x)}\,dx \\ &\qquad {} -C_{1} \bigl(1+ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}\bigr) \int _{\Sigma ^{c} \cap B_{r_{0}}} \vert z_{n} \vert ^{p(x)}\,dx -\widetilde{C}_{1} \\ &\quad \ge \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}V(x) \vert z_{n} \vert ^{p(x)}\,dx +\frac{1}{2} \biggl( \frac{1}{p^{+}}- \frac{1}{\mu } \biggr) \int _{\Sigma ^{c}\cap B_{r_{0}}^{c}}V(x) \vert z_{n} \vert ^{p(x)}\,dx \\ &\qquad {} -C_{1} \bigl(1+ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}\bigr) \int _{\Sigma ^{c} \cap B_{r_{0}}^{c}} \vert z_{n} \vert ^{p(x)}\,dx -\mathcal{M}_{1} \\ &\quad \ge \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}V(x) \vert z_{n} \vert ^{p(x)}\,dx -\mathcal{M}_{1}, \end{aligned}$$

where \(\mathcal{M}_{1}\) is a positive constant, as claimed. This fact, together with (F2) and (F4), leads to

$$\begin{aligned} &\mathcal{M}_{0}+o(1) \\ &\quad \ge \mathcal{I}_{\lambda }(z_{n})- \frac{1}{\mu } \bigl\langle \mathcal{I}_{\lambda }^{\prime }(z_{n}),z_{n} \bigr\rangle \\ &\quad = \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}}\frac{1}{p(x,y)} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy + \int _{\mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert z_{n} \rvert ^{p(x)}} \,dx \\ &\qquad {}-\lambda \int _{\mathbb{R}^{N}} \frac{a(x)}{r(x)} \vert z_{n} \vert ^{r(x)} \,dx - \int _{\mathbb{R}^{N}}F(x,z_{n}) \,dx \\ &\qquad {}-\frac{1}{\mu } \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy - \frac{1}{\mu } \int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx \\ &\qquad {} +\frac{\lambda }{\mu } \int _{\mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx+ \frac{1}{\mu } \int _{\mathbb{R}^{N}}f(x,z_{n})z_{n} \,dx \\ &\quad \ge \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy \\ &\qquad {}+ \biggl(\frac{1}{p^{+}}- \frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx -{\lambda } \biggl(\frac{1}{r_{-}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx \\ &\qquad {}- \int _{\mathbb{R}^{N}}F(x,z_{n}) \,dx+ \frac{1}{\mu } \int _{\mathbb{R}^{N}}f(x,z_{n})z_{n} \,dx \\ &\quad \geq \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy \\ &\qquad {}+ \biggl(\frac{1}{p^{+}}- \frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx -{\lambda } \biggl(\frac{1}{r_{-}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx \\ &\qquad {}+ \int _{ \vert z_{n} \vert > M} \biggl( \frac{1}{\mu }f(x, z_{n}) z_{n}- F(x, z_{n}) \biggr) \,dx - C_{1} \int _{ \vert z_{n} \vert \leq M}\bigl( \lvert z_{n} \rvert ^{p(x)}+b(x) \lvert z_{n} \rvert ^{q(x)}\bigr) \,dx \\ &\quad \geq \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy \\ &\qquad {}+ \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx -{\lambda } \biggl(\frac{1}{r_{-}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx \\ &\qquad {}-\frac{1}{\mu } \int _{\mathbb{R}^{N}} \bigl(\varrho (x) \lvert z_{n} \rvert ^{p^{-}}+\zeta (x) \bigr) \,dx-\mathcal{M}_{1} \\ &\quad \geq \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \biggl( \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy+ \int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx \biggr) \\ &\qquad {} -{\lambda } \biggl(\frac{1}{r_{-}}-\frac{1}{\mu } \biggr) \int _{ \mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx-\frac{1}{\mu } \int _{\mathbb{R}^{N}} \varrho (x) \lvert z_{n} \rvert ^{p^{-}} \,dx- \frac{1}{\mu } \Vert \zeta \Vert _{L^{1}(\mathbb{R}^{N})}-\mathcal{M}_{1} \\ &\quad \geq \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \Vert z_{n} \Vert _{ \mathcal{X}}^{p^{-}}- \biggl(\frac{{\lambda }}{r_{-}}- \frac{{\lambda }}{\mu } \biggr) \int _{\mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx \\ &\qquad {} -\frac{1}{\mu } \biggl( \int _{\mathcal{B}_{1}} \varrho (x) \lvert z_{n} \rvert ^{p^{-}} \,dx + \int _{ \mathcal{B}_{2}}\varrho (x) \lvert z_{n} \rvert ^{p^{-}} \,dx \biggr)-\frac{1}{\mu } \Vert \zeta \Vert _{L^{1}(\mathbb{R}^{N})}-\mathcal{M}_{1} \\ &\quad \geq \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \Vert z_{n} \Vert _{ \mathcal{X}}^{p^{-}}- \biggl(\frac{{\lambda }}{r_{-}}- \frac{{\lambda }}{\mu } \biggr) \int _{\mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx \\ &\qquad {} -\frac{1}{\mu } \biggl(2 \Vert \varrho \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-p^{-}}}(\mathcal{B}_{1})} \Vert z_{n} \Vert _{L^{p( \cdot )}(\mathcal{B}_{1})}^{p^{-}} +\tilde{ \varrho } \int _{ \mathcal{B}_{2}} \lvert z_{n} \rvert ^{p^{-}} \,dx \biggr)- \frac{1}{\mu } \Vert \zeta \Vert _{L^{1}(\mathbb{R}^{N})}-\mathcal{M}_{1} \\ &\quad \geq \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \Vert z_{n} \Vert _{ \mathcal{X}}^{p^{-}}- \biggl(\frac{{\lambda }}{r_{-}}- \frac{{\lambda }}{\mu } \biggr) \int _{\mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx \\ &\qquad {} -\frac{1}{\mu } \bigl(2 \Vert \varrho \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-p^{-}}}(\mathcal{B}_{1})} \Vert z_{n} \Vert _{L^{p( \cdot )}(\mathbb{R}^{N})}^{p^{-}} +\tilde{ \varrho } \Vert z_{n} \Vert _{L^{p( \cdot )}(\mathcal{B}_{2})}^{p^{-}} \bigr)-\frac{1}{\mu } \Vert \zeta \Vert _{L^{1}( \mathbb{R}^{N})}- \mathcal{M}_{1} \\ &\quad \geq \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \Vert z_{n} \Vert _{ \mathcal{X}}^{p^{-}}- \biggl(\frac{{\lambda }}{r_{-}}- \frac{{\lambda }}{\mu } \biggr) \int _{\mathbb{R}^{N}}a(x) \vert z_{n} \vert ^{r(x)} \,dx \\ &\qquad {} -\frac{1}{\mu } \bigl(2 \Vert \varrho \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-p^{-}}}(\mathcal{B}_{1})}+ \tilde{\varrho } \bigr) \Vert z_{n} \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{p^{-}}- \frac{1}{\mu } \Vert \zeta \Vert _{L^{1}(\mathbb{R}^{N})}- \mathcal{M}_{1} \\ &\quad \geq \frac{1}{2} \biggl(\frac{1}{p^{+}}-\frac{1}{\mu } \biggr) \Vert z_{n} \Vert _{ \mathcal{X}}^{p^{-}}- \biggl(\frac{{\lambda }}{r_{-}}- \frac{{\lambda }}{\mu } \biggr) \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})}\max \bigl\{ \Vert z_{n} \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{+}}, \Vert z_{n} \Vert _{L^{p(\cdot )}( \mathbb{R}^{N})}^{r_{-}}\bigr\} \\ &\qquad {} -\frac{1}{\mu } \bigl(2 \Vert \varrho \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-p^{-}}}(\mathcal{B}_{1})}+ \tilde{\varrho } \bigr) \Vert z_{n} \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{p^{-}}- \frac{1}{\mu } \Vert \zeta \Vert _{L^{1}(\mathbb{R}^{N})}- \mathcal{M}_{1} \end{aligned}$$

for sufficiently large n because \(\int _{\mathcal{B}_{2}} \lvert z_{n} \rvert ^{p^{-}} \,dx \le \int _{\mathbb{R}^{N}} \lvert z_{n} \rvert ^{p(x)} \,dx\). This fact implies

$$ \begin{aligned}[b] \frac{1}{2}&\leq \frac{p^{+} (2 \Vert \varrho \Vert _{L^{\frac{p(\cdot )}{p(\cdot )-p^{-}}}(\mathcal{B}_{1})}+\tilde{\varrho } )}{\mu -p^{+}} \limsup _{n\rightarrow \infty } \Vert w_{n} \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{p^{-}} \\ &= \frac{p^{+} (2 \Vert \varrho \Vert _{L^{\frac{p(\cdot )}{p(\cdot )-p^{-}}}(\mathcal{B}_{1})}+\tilde{\varrho } )}{\mu -p^{+}} \Vert \omega \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{p^{-}}. \end{aligned} $$
(3.3)

Hence, from (3.3), it follows that \(\omega \neq 0\). However, to obtain the boundedness of \(\{z_{n}\}\), we should prove that \(\omega (x)=0\) for almost all \(x \in \mathbb{R}^{N}\). Set \(\Omega _{1}= \{ x\in \mathbb{R}^{N} : \omega (x)\neq0 \} \). By virtue of relation (3.1), one has

$$\begin{aligned} \mathcal{I}_{\lambda }(z_{n})&= \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy + \int _{\mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert z_{n} \rvert ^{p(x)}} \,dx \\ &\quad {} -\lambda \int _{\mathbb{R}^{N}}\frac{a(x)}{r(x)} \vert z_{n} \vert ^{r(x)} \,dx- \int _{\mathbb{R}^{N}}F(x,z_{n}) \,dx \\ &\ge \frac{1}{p^{+}} \Vert z_{n} \Vert _{\mathcal{X}}^{p^{-}}- \frac{\lambda }{r_{-}} \Vert a \Vert _{L^{\frac{p(\cdot )}{p(\cdot )-r(\cdot )}}( \mathbb{R}^{N})}\max \bigl\{ \Vert z_{n} \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{+}}, \Vert z_{n} \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{-}}\bigr\} \\ &\quad {} - \int _{\mathbb{R}^{N}}F(x,z_{n})\,dx. \end{aligned}$$
(3.4)

Since \(\mathcal{I}_{\lambda }(z_{n})\leq \mathcal{M}_{0}\) for all \(n\in \mathbb{N}\) and \(\|z_{n}\|_{\mathcal{X}}\to \infty \) as \(n\to \infty \), we assert that

$$ \begin{aligned}[b] \int _{\mathbb{R}^{N}}F(x,z_{n})\,dx &\ge \frac{1}{ p^{+}} \Vert z_{n} \Vert _{ \mathcal{X}}^{p^{-}}- \frac{\lambda }{ r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})}\max \bigl\{ \Vert z_{n} \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{+}}, \Vert z_{n} \Vert _{L^{p(\cdot )}( \mathbb{R}^{N})}^{r_{-}}\bigr\} -{ \mathcal{M}}_{0} \\ &\to \infty \end{aligned} $$
(3.5)

as \(n\to \infty \). In addition,

$$\begin{aligned} \mathcal{I}_{\lambda }(z_{n})&= \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy + \int _{\mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert z_{n} \rvert ^{p(x)}} \,dx \\ &\quad {} -\lambda \int _{\mathbb{R}^{N}}\frac{a(x)}{r(x)} \vert z_{n} \vert ^{r(x)} \,dx- \int _{\mathbb{R}^{N}}F(x,z_{n}) \,dx \\ &\le \frac{1}{p^{-}} \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy \\ &\quad {}+ \frac{1}{p^{-}} \int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx- \int _{\mathbb{R}^{N}}F(x,z_{n}) \,dx. \end{aligned}$$

Then

$$ \begin{aligned}[b] &\frac{1}{p^{-}} \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}} \int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx \\ &\quad \geq \int _{\mathbb{R}^{N}}F(x,z_{n}) \,dx+\mathcal{I}_{\lambda }(z_{n}). \end{aligned} $$
(3.6)

In accordance with assumption (F3), there is a positive constant \(\tau _{0}>1\) such that \({F(x,\tau )}>{ \lvert \tau \rvert ^{p^{+}}}\) for all \(x\in \mathbb{R}^{N}\) and \(\lvert \tau \rvert >\tau _{0}\). From assumptions (F1) and (F2), it follows that there is \(\mathcal{M}_{2}>0\) such that \(\lvert F(x,\tau ) \rvert \leq \mathcal{M}_{2}\) for all \((x,\tau )\in \mathbb{R}^{N} \times [-\tau _{0},\tau _{0}]\). Therefore, we can choose a real number \(\mathcal{M}_{3}\) such that \(F(x,\tau )\geq \mathcal{M}_{3}\) for all \((x,\tau )\in \mathbb{R}^{N}\times \mathbb{R}\), and thus

$$ \frac{F(x,z_{n})-\mathcal{M}_{3}}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx} \geq 0, $$

for all \(x\in \mathbb{R}^{N}\) and for all \(n\in \mathbb{N}\). By convergence (3.2), we know that \(\lvert z_{n}(x) \rvert = \lvert w_{n}(x) \rvert \|z_{n}\|_{\mathcal{X}}\to \infty \) as \(n\to \infty \) for all \(x\in \Omega _{1}\). Furthermore, from assumption (F3) it follows that for all \(x\in \Omega _{1}\) we have

$$\begin{aligned} &\lim_{n\to \infty }{ \frac{F(x,z_{n})}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx}} \\ &\quad \geq \lim_{n\to \infty } \frac{F(x,z_{n})}{\frac{1}{p^{-}} \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}= \lim _{n\to \infty }{ \frac{p^{-} F(x,z_{n})}{ \lvert z_{n}(x) \rvert ^{p^{+}}} \bigl\lvert w_{n}(x) \bigr\rvert ^{p^{+}}}= \infty . \end{aligned}$$
(3.7)

Hence we infer that \(\lvert \Omega _{1} \rvert =0\). Indeed, if \(\lvert \Omega _{1} \rvert \neq0\), then, from relations (3.5)–(3.7) and invoking the Fatou lemma, it follows that

$$\begin{aligned} 1 &={\liminf_{n\to \infty }{ \frac{{\int _{\mathbb{R}^{N}}{F(x,z_{n})} \,dx}}{\int _{\mathbb{R}^{N}}{F(x,z_{n})} \,dx+\mathcal{I}_{\lambda }(z_{n})}}} \\ &\ge {\liminf_{n\to \infty } \int _{\mathbb{R}^{N}}{ \frac{F(x,z_{n})}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx}}} \,dx \\ & \ge {\liminf_{n\to \infty } \int _{\Omega _{1}}{ \frac{F(x,z_{n})}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx}}} \,dx \\ & - \limsup_{n\to \infty }{ \int _{\Omega _{1}} \frac{\mathcal{M}_{3}}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx} \,dx} \\ &= \liminf_{n\to \infty }{ \int _{\Omega _{1}}{ \frac{F(x,z_{n})-\mathcal{M}_{3}}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx}} \,dx} \\ &\ge \int _{\Omega _{1}}{\liminf_{n\to \infty }{ \frac{F(x,z_{n})-\mathcal{M}_{3}}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx}}} \,dx \\ & = \int _{\Omega _{1}}{\liminf_{n\to \infty }{ \frac{F(x,z_{n}(x))}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx}}} \,dx \\ & \quad {} -{ \int _{\Omega _{1}}\limsup_{n\to \infty }{ \frac{\mathcal{M}_{3}}{\frac{1}{p^{-}}\int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \vert z_{n}(x)-z_{n}(y) \vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \frac{1}{p^{-}}\int _{\mathbb{R}^{N}}{V(x) \lvert z_{n} \rvert ^{p(x)}} \,dx}} \,dx} \\ &=\infty . \end{aligned}$$

This is impossible. Thus we know \(\lvert \Omega _{1} \rvert =0\), and so \(\omega (x)=0\) for almost all \(x\in \mathbb{R}^{N}\), as claimed. Therefore we conclude that \(\{z_{n}\}\) is bounded in \(\mathcal{X}\). The proof is complete. □

Lemma 3.8

Assume conditions (H), (V), (A), (F1)(F3) and (F5). Furthermore, suppose that

  1. (F6)

    \(F(x,\tau )\ge 0\) for all \((x,\tau )\in \mathbb{R}^{N}\times \mathbb{R}^{+}\)

holds. Then

  1. (1)

    There is a positive constant \({\lambda }^{*}\) such that for any \({\lambda }\in (0,{\lambda }^{*})\) we can choose some constants \(R>0\) and \(0< r <1\) that \(\mathcal{I}_{\lambda }(z)\ge R>0\) for all \(z\in \mathcal{X}\) with \(\|z\|_{\mathcal{X}}=r\).

  2. (2)

    There exists \(z\in C_{c}^{\infty }(\mathbb{R}^{N})\), \(z>0\), such that \(\mathcal{I}_{\lambda }(tz)\to -\infty \) as \(t\to +\infty \).

  3. (3)

    There exists \(w\in C_{c}^{\infty }(\mathbb{R}^{N})\), \(w>0\), such that \(\mathcal{I}_{\lambda }(tw)<0\) for all \(t\to 0^{+}\).

Proof

Statement (1) is proved in [11, 37]. Thus, we first show statement (2). By assumptions (F2)–(F3) and (F5), for any \(\mathcal{M}>0\), there exist some constants \(C_{2}>0\) and \(C_{3}(\mathcal{M})>0\) such that

$$ F(x,\tau )\ge \mathcal{M} \lvert \tau \rvert ^{p^{+}}-C_{2} \vert \tau \vert ^{p(x)}-C_{3}( \mathcal{M})b(x) $$
(3.8)

for all \((x,\tau )\in \mathbb{R}^{N}\times \mathbb{R}\) where b comes from (F2). Let us take \(z\in C_{c}^{\infty }(\mathbb{R}^{N})\). Then relation (3.8) implies that

$$\begin{aligned} \mathcal{I}_{\lambda }(tz)&= \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert tz(x)-tz(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy+ \int _{ \mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert tz \rvert ^{p(x)}} \,dx \\ &\quad {} -\lambda \int _{\mathbb{R}^{N}}\frac{a(x)}{r(x)} \vert tz \vert ^{r(x)} \,dx- \int _{\mathbb{R}^{N}}F(x,tz) \,dx \\ &\le t^{p^{+}} \biggl( \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy+ \int _{ \mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert z \rvert ^{p(x)}} \,dx \\ & \quad {}-\mathcal{M} \int _{\mathbb{R}^{N}} \lvert z \rvert ^{p^{+}} \,dx +C_{2} \int _{\mathbb{R}^{N}} \vert z \vert ^{p(x)}\,dx \biggr) +C_{3} \end{aligned}$$

for \(t>1\) large enough and for a constant \(C_{3}\). If \(\mathcal{M}\) is sufficiently large, then we assert that \(\mathcal{I}_{\lambda }(tz)\to -\infty \) as \(t \to \infty \). Therefore the functional \(\mathcal{I}_{\lambda }\) is unbounded from below.

Next, we have to show (3). Let us choose \(w\in C_{c}^{\infty }(\mathbb{R}^{N})\) such that \(w>0\). For \(t>0\) small enough, from (A) and (F5), it follows that

$$\begin{aligned} \mathcal{I}_{\lambda }(tw)&= \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert tw(x)-tw(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy+ \int _{ \mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert tw \rvert ^{p(x)}} \,dx \\ &\quad {} -\lambda \int _{\mathbb{R}^{N}}\frac{a(x)}{r(x)} \vert tw \vert ^{r(x)} \,dx- \int _{\mathbb{R}^{N}}F(x,tw) \,dx \\ &\le t^{p^{-}} \biggl( \int _{\mathbb{R}^{N} \times \mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert w(x)-w(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy+ \int _{ \mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert w \rvert ^{p(x)}} \,dx \biggr) \\ &\quad {}-\frac{\lambda t^{r_{+}}}{r_{+}} \int _{\mathbb{R}^{N}}a(x) \vert w \vert ^{r(x)} \,dx. \end{aligned}$$

Since \(r_{+}< p^{-}\), it follows that \(\mathcal{I}_{\lambda }(tw)<0\) as \(t\to 0^{+}\). The proof is completed. □

The following lemma is the variational principle of Ekeland type in [6, 37], initially developed by C.-K. Zhong [56].

Lemma 3.9

([6, 37])

Let E be a Banach space and \(x_{0}\) be a fixed point of E. Suppose that \(g:{E}\rightarrow \mathbb{R}\cup \{+\infty \}\) is a lower semicontinuous function, not identically +∞, bounded from below. Then, for every \({\varepsilon }>0\) and \(y\in E\) such that

$$ g(y)< \inf_{{E}}{g}+{\varepsilon }$$

and every \(\lambda >0\), there exists some point \(z\in E\) such that

$$ g(z) \le g(y),\qquad \Vert z-x_{0} \Vert _{E}\le \bigl({1+ \Vert y \Vert _{E}}\bigr) \bigl(e^{{\lambda }}-1 \bigr), $$

and

$$ g(x)\ge g(z)-\frac{{\varepsilon }}{\lambda (1+ \Vert z \Vert _{E})}{ \Vert x-z \Vert _{E}} \quad \textit{for all } x\in E. $$

With the help of Lemmas 3.7, 3.8, and 3.9, we are in a position to derive our first main result.

Theorem 3.10

Assume that (H), (V), (A), and (F1)(F6) hold. Then there exists \({\lambda }^{*}>0\) such that, for any \({\lambda }\in (0,{\lambda }^{*})\), problem (P) admits at least two distinct nontrivial weak solutions.

Proof

By means of Lemmas 3.7 and 3.8, there is a positive real number \({\lambda }^{*}\) such that \(\mathcal{I}_{\lambda }\) ensures the mountain pass geometry and the \((C)\)-condition for any \({\lambda }\in (0,{\lambda }^{*})\). Thanks to the mountain pass theorem in [17], we deduce that \(\mathcal{I}_{\lambda }\) has a critical point \(z_{0}\in \mathcal{X}\) with \(\mathcal{I}_{\lambda }(z_{0})=\overline{c}>0=\mathcal{I}_{\lambda }(0)\). Thus problem (P) possesses a nontrivial weak solution \(z_{0}\).

Next we show the existence of the second weak solution of (P). Owing to Lemma 3.8, for fixed \(\lambda \in (0, \lambda ^{*} )\), there are positive constants R and \(r\in (0,1)\) such that \(\mathcal{I}_{{\lambda }}(z) \geq R >0\) for all \(u \in \mathcal{X}\) with \(\|z\|_{\mathcal{X}} = r\). Let us denote \(c:=\inf_{z\in \overline{B}_{r}}\mathcal{I}_{\lambda }(z)\) where \(B_{r}:=\{z\in \mathcal{X} : \|z\|_{\mathcal{X}}< r\}\) with a boundary \(\partial B_{r}\). Then, by Lemma 3.8(3), we know \(-\infty < c<0\). If we put \(0<{\varepsilon }<\inf_{z\in \partial {B}_{r}}\mathcal{I}_{\lambda }(z)-c\), from Lemma 3.9 it follows that we can look for \(z_{{\varepsilon }}\in \overline{B}_{r}\) such that

$$ \textstyle\begin{cases} \mathcal{I}_{\lambda }(z_{{\varepsilon }})< c+{\varepsilon }, \\ \mathcal{I}_{\lambda }(z_{{\varepsilon }})\leq \mathcal{I}_{\lambda }(z)+ \frac{{\varepsilon }}{1+ \Vert z_{{\varepsilon }} \Vert _{\mathcal{X}}} \Vert z-z_{{ \varepsilon }} \Vert _{\mathcal{X}} \quad \text{for all } z\in \overline{B}_{r} \text{ with } z\neq z_{{\varepsilon }}. \end{cases} $$
(3.9)

This fact together with the estimate \(\mathcal{I}_{\lambda }(z_{{\varepsilon }})< c+{\varepsilon }< \inf_{z \in \partial {B}_{r}}\mathcal{I}_{\lambda }(z)\) gives that \(z_{{\varepsilon }}\in B_{r}\). Hence it follows that \(z_{{\varepsilon }}\) is a local minimum of \(\widetilde{I}_{{\lambda }}(z):=\mathcal{I}_{\lambda }(z)+ \frac{{\varepsilon }}{1+\|z_{{\varepsilon }}\|_{\mathcal{X}}}\|z-z_{{ \varepsilon }}\|_{\mathcal{X}}\). Now, by taking \(z=z_{{\varepsilon }}+t\omega \) for \(\omega \in B_{1}\) and \(t>0\) small enough, we deduce from (3.9) that

$$ 0\leq \frac{\widetilde{I}_{{\lambda }}(z_{{\varepsilon }}+t\omega ) -\widetilde{I}_{{\lambda }}(z_{{\varepsilon }})}{t}= \frac{\mathcal{I}_{{\lambda }}(z_{{\varepsilon }}+t\omega ) -\mathcal{I}_{{\lambda }}(z_{{\varepsilon }})}{t}+ \frac{{\varepsilon }}{1+ \Vert z_{{\varepsilon }} \Vert _{\mathcal{X}}} \Vert \omega \Vert _{\mathcal{X}}. $$

Therefore, letting \(t\to 0^{+}\), we get

$$ \bigl\langle \mathcal{I}_{\lambda }'(z_{{\varepsilon }}), \omega \bigr\rangle + \frac{{\varepsilon }}{1+ \Vert z_{{\varepsilon }} \Vert _{\mathcal{X}}} \Vert \omega \Vert _{\mathcal{X}} \ge 0. $$

Changing ω into −ω in the argument above, one has

$$ -\bigl\langle \mathcal{I}_{\lambda }'(z_{{\varepsilon }}), \omega \bigr\rangle + \frac{{\varepsilon }}{1+ \Vert z_{{\varepsilon }} \Vert _{\mathcal{X}}} \Vert \omega \Vert _{\mathcal{X}} \ge 0. $$

Thus, we have

$$ \bigl(1+ \Vert z_{{\varepsilon }} \Vert _{\mathcal{X}}\bigr) \bigl\vert \bigl\langle \mathcal{I}_{\lambda }'(z_{{ \varepsilon }}), \omega \bigr\rangle \bigr\vert \le {\varepsilon } \Vert \omega \Vert _{\mathcal{X}} $$

for any \(\omega \in \overline{B}_{1}\). Hence we know

$$ \bigl(1+ \Vert z_{{\varepsilon }} \Vert _{\mathcal{X}} \bigr) \bigl\Vert \mathcal{I}_{\lambda }'(z_{{ \varepsilon }}) \bigr\Vert _{\mathcal{X}^{*}} \le {\varepsilon }. $$
(3.10)

Using (3.9) and (3.10), we can choose a sequence \(\{z_{n}\}\subset B_{r}\) such that

$$ \textstyle\begin{cases} \mathcal{I}_{\lambda }(z_{n})\to c \quad \text{as } n\to \infty, \\ (1+ \Vert z_{{\varepsilon }} \Vert _{\mathcal{X}}) \Vert \mathcal{I}_{\lambda }'(z_{{ \varepsilon }}) \Vert _{\mathcal{X}^{*}}\to 0 \quad \text{as } n\to \infty . \end{cases} $$
(3.11)

Thus, \(\{z_{n}\}\) is a bounded \((C)\)-sequence in the reflexive Banach space \(\mathcal{X}\). By virtue of the fact that \(\mathcal{I}_{\lambda }'\) is of type \((S_{+})\) as mentioned in the proof of Lemma 3.7, \(\{z_{n}\}\) has a subsequence \(\{z_{n_{k}}\}\) such that \(z_{n_{k}}\to z_{1}\) in \(\mathcal{X}\) as \(k\to \infty \). This fact together with (3.11) leads to \(\mathcal{I}_{\lambda }(z_{1})=c<0\) and \(\mathcal{I}_{\lambda }'(z_{1})=0\). Hence there is a nontrivial solution \(z_{1}\) which is different from \(z_{0}\). Therefore we conclude that problem (P) possesses at least two distinct nontrivial weak solutions. □

Next, by applying the fountain theorem and the dual fountain theorem as essential tools which are originally provided by the papers [9] and [10], we establish two existence results of a sequence of infinitely many solutions for problem (P). Let E be a real reflexive and separable Banach space, then it is known (see [22, 57]) that there exist \(\{e_{n}\}\subseteq W\) and \(\{f_{n}^{*}\}\subseteq E^{*}\) such that

$$ E=\overline{\operatorname{span}\{e_{n}:n=1,2,\ldots \}},\qquad E^{*}=\overline{\operatorname{span}\bigl\{ f_{n}^{*}:n=1,2, \ldots \bigr\} }, $$

and

$$ \bigl\langle f^{*}_{i},e_{j} \bigr\rangle = \textstyle\begin{cases} 1 &\text{if } i=j, \\ 0 &\text{if } i\ne j. \end{cases} $$

Let us denote \(E_{n}=\operatorname{span}\{e_{n}\}\), \(Y_{k}=\bigoplus_{n=1}^{k}E_{n}\), and \(Z_{k}= \overline{\bigoplus_{n=k}^{\infty }E_{n}}\).

Lemma 3.11

(Fountain theorem [9, 30, 52])

Let E be a Banach space, the functional \(\mathcal{I} \in C^{1}(E,\mathbb{R})\) satisfies the \((C)_{c}\)-condition for any \(c>0\) and \(\mathcal{I}\) is even. If for each sufficiently large \(k \in \mathbb{N}\) there exist \(\rho _{k}> \delta _{k}>0\) such that the following properties hold:

  1. (1)

    \(b_{k}:=\inf \{ \mathcal{I}(z):z\in Z_{k}, \|z\|_{E}= \delta _{k}\}\to \infty\) as \(k\to \infty \);

  2. (2)

    \(a_{k}:=\max \{ \mathcal{I}(z):z\in Y_{k}, \|z\|_{E}= \rho _{k}\}\le 0\),

then \(\mathcal{I}\) possesses an unbounded sequence of critical values, i.e., there is a sequence \(\{z_{n}\}\subset X\) such that \(\mathcal{I}^{\prime }(z_{n})=0\) and \(\mathcal{I}(z_{n})\to +\infty \) as \(n\to +\infty \).

With the aid of Lemma 3.11, we are in a position to derive the existence of multiple large energy solutions.

Theorem 3.12

Assume that (H), (V), (A), and (F1)(F4) hold. If \(f(x,-t)=-f(x,t)\) holds for all \((x,t)\in \mathbb{R}^{N} \times \mathbb{R}\), then for any \(\lambda >0\) problem (P) possesses a sequence of nontrivial weak solutions \(\{z_{n}\}\) in \(\mathcal{X}\) such that \(\mathcal{I}_{\lambda }( z_{n})\to \infty \) as \(n\to \infty \).

Proof

Clearly, \(\mathcal{I}_{\lambda }\) is an even functional that ensures the \((C)_{c}\)-condition. It is enough to prove that there are \(\rho _{k}> \delta _{k}>0\) such that

  1. (1)

    \(b_{k}:=\inf \{\mathcal{I}_{\lambda }(z):z\in Z_{k}, \|z \|_{\mathcal{X}}= \delta _{k}\}\to \infty\) as \(k\to \infty \);

  2. (2)

    \(a_{k}:=\max \{\mathcal{I}_{\lambda }(z):z\in Y_{k}, \|z \|_{\mathcal{X}}=\rho _{k}\}\le 0\),

for sufficiently large k. Denote

$$ \alpha _{k}:=\sup_{z \in Z_{k}, \Vert z \Vert _{\mathcal{X}}=1} \Vert z \Vert _{L^{q( \cdot )}(\mathbb{R}^{N})}. $$

Then we assert \(\alpha _{k} \to 0\) as \(k \to \infty \). Indeed, suppose to the contrary that we can choose \(\varepsilon _{0}>0\), \(k_{0}\ge 0\), and the sequence \(\{z_{k}\}\) in \(Z_{k}\) such that

$$ \Vert z_{k} \Vert _{\mathcal{X}}=1,\qquad \Vert z \Vert _{L^{q(\cdot )}(\mathbb{R}^{N})} \ge \varepsilon _{0} $$

for all \(k \ge k_{0}\). From the boundedness of the sequence \(\{z_{k}\}\) in \(\mathcal{X}\), we look for \(z \in \mathcal{X}\) such that \(z_{k} \rightharpoonup z\) in \(\mathcal{X}\) as \(n\to \infty \) and

$$ \bigl\langle {f_{j}^{*},z}\bigr\rangle =\lim _{k \to \infty }{\bigl\langle {f_{j}^{*},z_{k}} \bigr\rangle }=0 $$

for \(j=1,2,\ldots \) . Hence we get \(z=0\). However, we obtain

$$ \varepsilon _{0} \le \lim_{k \to \infty }{ \Vert z_{k} \Vert _{L^{q(\cdot )}( \mathbb{R}^{N})}}= \Vert z \Vert _{L^{q(\cdot )}(\mathbb{R}^{N})}=0, $$

that is a contradiction.

For any \(z \in Z_{k}\), suppose that \(\|z\|_{\mathcal{X}}>1\). From (F2), Lemma 2.1, and (3.4), it follows that

$$\begin{aligned} \mathcal{I}_{\lambda }(z) &= \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert z(x)-z(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy + \int _{\mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert z \rvert ^{p(x)}} \,dx \\ &\quad {} -\lambda \int _{\mathbb{R}^{N}}\frac{a(x)}{r(x)} \vert z \vert ^{r(x)} \,dx- \int _{\mathbb{R}^{N}}F(x,z) \,dx \\ &\ge \frac{1}{p^{+}} \Vert z \Vert _{\mathcal{X}}^{p^{-}}- \frac{\lambda }{r_{-}} \Vert a \Vert _{L^{\frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} \max \bigl\{ \Vert z \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{+}}, \Vert z \Vert _{L^{p( \cdot )}(\mathbb{R}^{N})}^{r_{-}}\bigr\} \\ &\quad {} - \int _{\mathbb{R}^{N}} \frac{ \lvert b(x) \rvert }{q(x)} \lvert z \rvert ^{q(x)} \,dx \\ &\ge \frac{1}{p^{+}} \Vert z \Vert _{\mathcal{X}}^{p^{-}}- \frac{\lambda }{r_{-}} \Vert a \Vert _{L^{\frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} \max \bigl\{ \Vert z \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{+}}, \Vert z \Vert _{L^{p( \cdot )}(\mathbb{R}^{N})}^{r_{-}}\bigr\} \\ &\quad {}- \frac{ \Vert b \Vert _{L^{\infty } (\mathbb{R}^{N})}}{q_{-}} \int _{\mathbb{R}^{N}} \lvert z \rvert ^{q(x)} \,dx \\ &\ge \frac{1}{p^{+}} \Vert z \Vert _{\mathcal{X}}^{p^{-}}- \frac{\lambda }{r_{-}}C_{4} \Vert z \Vert _{\mathcal{X}}^{r_{+}} -\frac{1}{q_{-}}\alpha _{k}^{q_{-}}C_{5} \Vert z \Vert ^{q_{+}}_{\mathcal{X}}, \end{aligned}$$

where \(C_{4}\) and \(C_{5}\) are positive constants. Now, let us choose \(\delta _{k}=( q_{+}C_{5}\alpha _{k}^{q_{-}}/{q_{-}})^{{1}/ (p^{-}-q_{+} )}\). Since \(p^{-}< q_{+}\) and \(\alpha _{k}\to 0\) as \(k\to \infty \), we assert \(\delta _{k}\to \infty \) as \(k\to \infty \). Hence, if \(z \in Z_{k}\) and \(\|z\|_{\mathcal{X}}=\delta _{k}\), then we arrive at

$$ \mathcal{I}_{\lambda }(z)\ge \biggl(\frac{1}{p^{+}}- \frac{1}{q_{+}} \biggr)\delta _{k}^{p^{-}}- \frac{\lambda }{r_{-}}C_{4} \delta _{k}^{r_{+}} \to \infty \quad \text{as } k\to \infty , $$

which implies condition (1).

Now we prove condition (2) arguing by contradiction. Then, let us assume that condition (2) is not satisfied for some k. Then we can find a sequence \(\{z_{n}\}\) in \(Y_{k}\) such that

$$ \Vert z_{n} \Vert _{\mathcal{X}}\to \infty \quad \text{as } n\to \infty\quad \text{and} \quad I_{\lambda }(z_{n}) \ge 0. $$
(3.12)

Let \(w_{n}=z_{n}/\|z_{n}\|_{\mathcal{X}}\). Then, clearly, we have \(\|w_{n}\|_{\mathcal{X}}=1\). Since \(\dim {Y_{k}}<\infty \), there is an element w in \(Y_{k}\setminus \{0\}\) such that, up to a subsequence still denoted by \(\{w_{n}\}\),

$$ \Vert w_{n}-w \Vert _{\mathcal{X}}\to 0 \quad \text{and}\quad w_{n}(x)\to w(x) $$

for almost all \(x\in \mathbb{R}^{N}\) as \(n\to \infty \). We claim that \(w(x)=0\) for almost all \(x\in \mathbb{R}^{N}\). If \(w(x)\neq 0\), then \(\lvert z_{n}(x) \rvert \to \infty \) for all \(x\in \mathbb{R}^{N}\) as \(n\to \infty \). Hence, by means of assumption (F3) it follows that

$$ \lim_{n\to \infty }{ \frac{F(x,z_{n}(x))}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}} = \lim _{n\to \infty }{ \frac{F(x,z_{n}(x))}{ \lvert z_{n}(x) \rvert ^{p^{+}}} \bigl\lvert w_{n}(x) \bigr\rvert ^{p^{+}} } = \infty $$
(3.13)

for all \(x\in \Omega _{2}:= \{ x\in \mathbb{R}^{N} : w(x)\neq0 \} \). Proceeding as in the proof of Lemma 3.7, it can be shown that there is \(\mathcal{M}_{2}\in \mathbb{R}\) such that \(F(x,t)\ge \mathcal{M}_{2}\) for all \((x,t)\in \mathbb{R}^{N}\times \mathbb{R}\), and so

$$ \frac{F(x,z_{n})-\mathcal{M}_{2}}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}} \ge 0 $$

for all \(x\in \mathbb{R}^{N}\) and \(n\in \mathbb{N}\). Using (3.13) and the Fatou lemma, one has

$$\begin{aligned} \liminf_{n\to \infty } \int _{\mathbb{R}^{N}} \frac{F(x,z_{n})}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}\,dx &\ge \liminf _{n \to \infty } \int _{\Omega _{2}} \frac{F(x,z_{n})}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}\,dx -\limsup _{n\to \infty } \int _{\Omega _{2}} \frac{\mathcal{M}_{2}}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}\,dx \\ &= \liminf_{n\to \infty } \int _{\Omega _{2}} \frac{F(x,z_{n})-\mathcal{M}_{2}}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}\,dx \\ &\ge \int _{\Omega _{2}}\liminf_{n\to \infty } \frac{F(x,z_{n})-\mathcal{M}_{2}}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}\,dx \\ &= \int _{\Omega _{2}}\liminf_{n\to \infty } \frac{F(x,z_{n})}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}\,dx - \int _{\Omega _{2}} \limsup_{n\to \infty } \frac{\mathcal{M}_{2}}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}\,dx. \end{aligned}$$

Thus we infer

$$ \int _{\mathbb{R}^{N}}{ \frac{F(x,z_{n}(x))}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}} \,dx\to \infty \quad \text{as }n\to \infty . $$

We may assume that \(\|z_{n}\|_{\mathcal{X}}>1\). Therefore, we have

$$\begin{aligned} \mathcal{I}_{\lambda }(z_{n})&= \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert z_{n}(x)-z_{n}(y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy + \int _{\mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert z_{n} \rvert ^{p(x)}} \,dx \\ &\quad {} -\lambda \int _{\mathbb{R}^{N}}\frac{a(x)}{r(x)} \vert z_{n} \vert ^{r(x)} \,dx- \int _{\mathbb{R}^{N}}F(x,z_{n}) \,dx \\ &\le \frac{1}{p^{-}} \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}- \int _{\Omega _{2}}{F(x,z_{n})} \,dx \\ &\le \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}} \biggl(\frac{1}{p^{-}}- \int _{ \Omega _{2}}{\frac{F(x,z_{n}(x))}{ \Vert z_{n} \Vert _{\mathcal{X}}^{p^{+}}}} \,dx \biggr)\to -\infty \quad \text{as }n\to \infty , \end{aligned}$$

which is a contradiction to (3.12). This completes the proof. □

Definition 3.13

Let E be a real separable and reflexive Banach space. We say that \(\mathcal{I}\) satisfies the \((C)_{c}^{*}\)-condition (with respect to \(Y_{n}\)) if any sequence \(\{z_{n}\}_{n\in \mathbb{N}}\subset E\) for which \(z_{n}\in Y_{n}\), for any \(n\in \mathbb{N}\),

$$ \mathcal{I}(z_{n}) \to c \quad \text{and}\quad \bigl\Vert (\mathcal{I}|_{Y_{n}})^{ \prime }(z_{n}) \bigr\Vert _{E^{*}}\bigl(1+ \Vert z_{n} \Vert _{E}\bigr) \to 0\quad \text{as } n\to \infty , $$

has a subsequence converging to a critical point of \(\mathcal{I}\).

Lemma 3.14

(Dual fountain theorem [10, 30])

Assume that E is a Banach space, \(\mathcal{I} \in C^{1}(E,\mathbb{R})\) is an even functional. If there is \(k_{0}>0\) so that, for each \(k\ge k_{0}\), there exist \(\rho _{k}> \delta _{k}>0\) such that the following properties hold:

  1. (H1)

    \(\inf \{\mathcal{I}(\omega ) : \omega \in Z_{k}, \| \omega \|_{E}=\rho _{k}\}\ge 0\);

  2. (H2)

    \(b_{k}:=\max \{ \mathcal{I}(\omega ):\omega \in Y_{k}, \|\omega \|_{E}=\delta _{k}\}< 0\);

  3. (H3)

    \(d_{k}:=\inf \{ \mathcal{I}(\omega ):\omega \in Z_{k}, \|\omega \|_{E}\le \rho _{k}\}\to 0\) as \(k\to \infty \);

  4. (H4)

    \(\mathcal{I}\) satisfies the \((C)_{c}^{*}\)-condition for every \(c\in [d_{k_{0}},0)\),

then \(\mathcal{I}\) has a sequence of negative critical values \(c_{n}<0\) satisfying \(c_{n}\to 0\) as \(n\to \infty \).

Lemma 3.15

Suppose that (H), (V), (A), and (F1)(F5) hold. Then the functional \(\mathcal{I}_{{\lambda }}\) satisfies the \((C)_{c}^{*}\)-condition for any \(\lambda >0\).

Proof

Since \(\mathcal{X}\) is a reflexive Banach space, and \(\mathcal{A}'\) and \(\Psi '_{{\lambda }}\) are of type \((S_{+})\), the proof is almost identical to that of Lemma 3.12 in [30]. □

With the help of Lemmas 3.14 and 3.15 we are ready to establish our final consequence.

Theorem 3.16

Assume that (H), (V), (A), and (F1)(F5) hold. Then problem (P) admits a sequence of nontrivial weak solutions \(\{\omega _{n}\}\) in \(\mathcal{X}\) such that \(\mathcal{I}_{{\lambda }}(\omega _{n})\to 0\) as \(n\to \infty \) for any \(\lambda >0\).

Proof

By means of (F4) and Lemma 3.15, we infer that the functional \(\mathcal{I}_{{\lambda }}\) is even and ensures the \((C)_{c}^{*}\)-condition for all \(c\in \mathbb{R}\). Now we will prove that properties (H1), (H2), and (H3) of the dual fountain theorem hold.

(H1): In accordance with (F1), we have

$$ \bigl\vert F(x,\tau ) \bigr\vert \le \frac{b(x)}{q(x)} \vert \tau \vert ^{q(x)}, \quad (x,\tau )\in \mathbb{R}^{N}\times \mathbb{R}.$$

For convenience, we denote

$$ \theta _{1,k}=\sup_{ \Vert \omega \Vert _{\mathcal{X}}=1,\omega \in Z_{k}} \Vert \omega \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})},\qquad \theta _{2,k}=\sup_{ \Vert \omega \Vert _{\mathcal{X}}=1,\omega \in Z_{k}} \Vert \omega \Vert _{L^{q(\cdot )}( \mathbb{R}^{N})}.$$

Then it is easy to verify that \(\theta _{1,k}\to 0\) and \(\theta _{2,k}\to 0\) as \(k\to \infty \) (see [30]). Set \(\vartheta _{k}=\max \{\theta _{1,k}, \theta _{2,k}\}\). Then it follows that

$$\begin{aligned} \mathcal{I}_{\lambda }(\omega ) &= \int _{\mathbb{R}^{N}} \int _{\mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert \omega (x)- \omega (y) \bigr\vert ^{p(x,y)}\mathcal{K}(x,y) \,dx\,dy + \int _{\mathbb{R}^{N}}{ \frac{V(x)}{p(x)} \lvert \omega \rvert ^{p(x)}} \,dx \\ &\quad {} -\lambda \int _{\mathbb{R}^{N}}\frac{a(x)}{r(x)} \vert \omega \vert ^{r(x)} \,dx- \int _{\mathbb{R}^{N}}F(x,\omega ) \,dx \\ &\ge \frac{1}{p^{+}} \Vert \omega \Vert _{\mathcal{X}}^{p^{-}}- \frac{\lambda }{r_{-}} \Vert a \Vert _{L^{\frac{p(\cdot )}{p(\cdot )-r(\cdot )}}( \mathbb{R}^{N})}\max \bigl\{ \Vert \omega \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{+}}, \Vert \omega \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{-}}\bigr\} \\ &\quad {} - \frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}}\max \bigl\{ \Vert \omega \Vert _{L^{q( \cdot )}(\mathbb{R}^{N})}^{q_{-}}, \Vert \omega \Vert _{L^{q(\cdot )}( \mathbb{R}^{N})}^{q_{+}} \bigr\} \\ &\ge \frac{1}{p^{+}} \Vert \omega \Vert _{\mathcal{X}}^{p^{-}}- \frac{\lambda }{r_{-}} \Vert a \Vert _{L^{\frac{p(\cdot )}{p(\cdot )-r(\cdot )}}( \mathbb{R}^{N})}\vartheta _{1,k}^{r_{-}} \Vert \omega \Vert _{\mathcal{X}}^{r_{+}} -\frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}}\vartheta _{2,k}^{q_{-}} \Vert \omega \Vert _{\mathcal{X}}^{q_{+}} \\ &\ge \frac{1}{p^{+}} \Vert \omega \Vert _{\mathcal{X}}^{p^{-}}- \biggl( \frac{2{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} + \frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}} \biggr)\vartheta _{k}^{r_{-}} \Vert \omega \Vert _{\mathcal{X}}^{q_{+}} \end{aligned}$$

for k large enough and \(\|\omega \|_{\mathcal{X}}\ge 1\). Choose

$$ \rho _{k}= \biggl[ \biggl(\frac{4{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} + \frac{2 \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}} \biggr)p^{+} \vartheta _{k}^{r_{-}} \biggr]^{\frac{1}{p^{-}-2q_{+}}}.$$

Let \(\omega \in Z_{k}\) with \(\|\omega \|_{\mathcal{X}}=\rho _{k}>1\) for sufficiently large k. Then there is \(k_{0}\in \mathbb{N}\) such that

$$\begin{aligned} \mathcal{I}_{{\lambda }}(\omega )&\ge \frac{1}{p^{+}} \Vert \omega \Vert _{ \mathcal{X}}^{p^{-}} - \biggl(\frac{2{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} + \frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}} \biggr)\vartheta _{k}^{r_{-}} \Vert \omega \Vert _{\mathcal{X}}^{2q_{+}} \\ &\ge \frac{1}{2p^{+}}\rho _{k}^{p^{-}}\ge 0 \end{aligned}$$

for all \(k\in \mathbb{N}\) with \(k\ge k_{0}\), being

$$ \lim_{k\to \infty }\frac{1}{2p^{+}}\rho _{k}^{p^{-}} =\infty . $$

Therefore,

$$ \inf \bigl\{ \mathcal{I}_{{\lambda }}(\omega ):\omega \in Z_{k}, \Vert \omega \Vert _{\mathcal{X}}=\rho _{k}\bigr\} \ge 0.$$

(H2): Observe that \(\|\cdot \|_{L^{p(\cdot )}(\mathbb{R}^{N})}\), \(\|\cdot \|_{L^{p^{+}}(\mathbb{R}^{N})}\) and \(\|\cdot \|_{\mathcal{X}}\) are equivalent on \(Y_{k}\). Then we can choose some constants \(\varsigma _{1,k}>0\) and \(\varsigma _{2,k}>0\) such that

$$ \Vert \omega \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}\le \varsigma _{1,k} \Vert \omega \Vert _{\mathcal{X}}\quad \text{and}\quad \Vert \omega \Vert _{\mathcal{X}}\le \varsigma _{2,k} \Vert \omega \Vert _{L^{p^{+}}(\mathbb{R}^{N})} $$
(3.14)

for any \(\omega \in Y_{k}\). From (F2)–(F3) and (F5), for any \(\mathcal{M}>0\), there are some constants \(C_{6}>0\) and \(C_{7}(\mathcal{M})>0\) such that

$$ F(x,\tau )\ge \mathcal{M}\varsigma _{2,k}^{p^{+}} \vert \tau \vert ^{p^{+}} -C_{6} \vert \tau \vert ^{p(x)} -C_{7}(\mathcal{M})b(x) $$
(3.15)

for almost all \((x,\tau )\in \mathbb{R}^{N}\times \mathbb{R}\) where b comes from (H2). Then, from (3.14) and (3.15), it follows that

$$\begin{aligned} \mathcal{I}_{{\lambda }}(\omega ) &\le \int _{\mathbb{R}^{N}} \int _{ \mathbb{R}^{N}} \frac{1}{p(x,y)} \bigl\vert \omega (x)- \omega (y) \bigr\vert ^{p(x,y)} \mathcal{K}(x,y) \,dx\,dy \\ &\quad {}+ \int _{\mathbb{R}^{N}}{\frac{V(x)}{p(x)} \lvert \omega \rvert ^{p(x)}} \,dx - \int _{\mathbb{R}^{N}}F(x, \omega ) \,dx \\ &\le \frac{1}{p^{-}} \Vert \omega \Vert _{\mathcal{X}}^{p^{+}} -\mathcal{M} \varsigma _{2,k}^{p^{+}} \int _{\mathbb{R}^{N}} \vert \omega \vert ^{p^{+}}\,dx +C_{6} \int _{\mathbb{R}^{N}} \vert \omega \vert ^{p(x)} +C_{7}(\mathcal{M}) \int _{ \mathbb{R}^{N}}b(x)\,dx \\ &\le \frac{1}{p^{-}} \Vert \omega \Vert _{\mathcal{X}}^{p^{+}} -\mathcal{M} \Vert \omega \Vert _{\mathcal{X}}^{p^{+}} +C_{6}\bigl(\varsigma _{1,k}^{p^{+}}+ \varsigma _{1,k}^{p^{-}}\bigr) \Vert \omega \Vert _{\mathcal{X}}^{p^{+}}+C_{8} \end{aligned}$$

for any \(\omega \in Y_{k}\) with \(\|\omega \|_{\mathcal{X}}\ge 1\) and positive constant \(C_{8}\). Let \(f(t)=\frac{1}{p^{-}}t^{p^{+}}-\mathcal{M}t^{p^{+}}+C_{6}(\varsigma _{1,k}^{p^{+}}+ \varsigma _{1,k}^{p^{-}})t^{p^{+}}+C_{8}\). If \(\mathcal{M}\) is large enough, then \(\lim_{t\to \infty }f(t)=-\infty \), and so there is \(t_{0}\in (1,\infty )\) such that \(f(t)<0\) for all \(t\in [t_{0},\infty )\). Hence \(\mathcal{I}_{{\lambda }}(\omega )<0\) for all \(\omega \in Y_{k}\) with \(\|\omega \|_{\mathcal{X}}=t_{0}\). Choosing \(\delta _{k}=t_{0}\) for all \(k\in \mathbb{N}\), one has

$$ b_{k}:=\max \bigl\{ \mathcal{J}_{{\lambda }}(\omega ):\omega \in Y_{k}, \Vert \omega \Vert _{\mathcal{X}}=\delta _{k}\bigr\} < 0.$$

If necessary, we can change \(k_{0}\) to a large value, so that \(\rho _{k}>\delta _{k}>0\) for all \(k\ge k_{0}\).

(H3): Because \(Y_{k}\cap Z_{k}\ne \emptyset \) and \(0<\delta _{k}<\rho _{k}\), we have \(d_{k}\le b_{k}<0\) for all \(k\ge k_{0}\). For any \(\omega \in Z_{k}\) with \(\|\omega \|_{\mathcal{X}}=1\) and \(0< t<\rho _{k}\), one has

$$\begin{aligned} \mathcal{I}_{{\lambda }}(t\omega )&\ge \frac{1}{p^{+}} \Vert t\omega \Vert _{ \mathcal{X}}^{p^{-}}-\frac{2{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} \max \bigl\{ \Vert t\omega \Vert _{L^{p(\cdot )}(\mathbb{R}^{N})}^{r_{-}}, \Vert t\omega \Vert _{L^{p( \cdot )}(\mathbb{R}^{N})}^{r_{+}} \bigr\} \\ &\quad {} -\frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}}\max \bigl\{ \Vert t \omega \Vert _{L^{q(\cdot )}(\mathbb{R}^{N})}^{q_{-}}, \Vert t\omega \Vert _{L^{q( \cdot )}(\mathbb{R}^{N})}^{q_{+}} \bigr\} \\ &\ge -\frac{2{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} \rho _{k}^{r_{+}} \vartheta _{k}^{r_{-}} - \frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}}\rho _{k}^{q_{+}} \vartheta _{k}^{q_{-}} \end{aligned}$$

for large enough k. Hence, from the definition of \(\rho _{k}\), it follows that

$$\begin{aligned} d_{k}&\ge -\frac{2{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} \rho _{k}^{r_{+}} \vartheta _{k}^{r_{-}} - \frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}}\rho _{k}^{q_{+}} \vartheta _{k}^{q_{-}} \\ &= -\frac{2{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} \biggl[{p^{+}} \biggl(\frac{4{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} + \frac{2 \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}} \biggr) \biggr]^{ \frac{r_{+}}{p^{-}-2q_{+}}} \vartheta _{k}^{ \frac{r_{-}r_{+}+(p^{-}-2q_{+})r_{-}}{p^{-}-2q_{+}}} \\ &\quad {} -\frac{ \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}} \biggl[{p^{+}} \biggl( \frac{4{\lambda }}{r_{-}} \Vert a \Vert _{L^{ \frac{p(\cdot )}{p(\cdot )-r(\cdot )}}(\mathbb{R}^{N})} + \frac{2 \Vert b \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q_{-}} \biggr) \biggr]^{ \frac{q_{+}}{p^{-}-2q_{+}}} \vartheta _{k}^{ \frac{r_{-}q_{+}+(p^{-}-2q_{+})q_{-}}{p^{-}-2q_{+}}}. \end{aligned}$$

Since \(p^{-}< q_{+}\), \(r_{+}+p^{-}<2q_{+}\), \(r_{-}q_{+}+q_{-}p^{-}<2q_{-}q_{+}\), and \(\vartheta _{k}\to 0\) as \(k\to \infty \), we arrive at \(\lim_{k\to \infty }d_{k}=0\).

Then, all the properties of Lemma 3.14 are satisfied. Consequently we conclude that problem (P) admits a sequence of nontrivial weak solutions \(\{\omega _{n}\}\) in \(\mathcal{X}\) such that \(\mathcal{I}_{{\lambda }}(\omega _{n})\to 0\) as \(n\to \infty \) for any \(\lambda >0\). □

Remark 3.17

In order to obtain a result similar to Theorem 3.16, the authors in [10, 41, 51, 52] have applied the dual fountain theorem when \(\rho _{k}\)—defined in Lemma 3.14—converges to 0 as \(k\to \infty \). For this reason, the proof of Theorem 3.16 is different from that of the papers [10, 41, 51, 52] because we get this result when \(\rho _{k}\to \infty \) as \(k\to \infty \).

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References

  1. Alves, C.O., Liu, S.B.: On superlinear \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 73, 2566–2579 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  Google Scholar 

  4. Azroul, E., Benkirane, A., Chung, N.T., Shimi, M.: Existence results for anisotropic fractional \((p_{1}(x,\cdot),p_{2}(x,\cdot))\)-Kirchhoff type problems. J. Appl. Anal. Comput. 11, 2363–2386 (2021)

    MathSciNet  Google Scholar 

  5. Azroul, E., Boumazourh, A., Chung, N.T.: Existence of solutions for a class of fractional Kirchhoff-type systems in \(\mathbb{R}^{N}\) with non-standard growth. Preprint

  6. Bae, J.-H., Kim, Y.-H.: Critical points theorems via the generalized Ekeland variational principle and its application to equations of \(p(x)\)-Laplace type in \(\mathbb{R}^{N}\). Taiwan. J. Math. 23, 193–229 (2019)

    MATH  Google Scholar 

  7. Bahrouni, A.: Comparison and sub-supersolution principles for the fractional \(p(x)\)-Laplacian. J. Math. Anal. Appl. 458, 1363–1372 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Bahrouni, A., Rădulescu, V.: On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin. Dyn. Syst., Ser. S 11, 379–389 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. 20, 1205–1216 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Bartsch, T., Willem, M.: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123, 3555–3561 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Biswas, R., Tiwari, S.: Existence, multiplicity and regularity of solutions of elliptic problem involving non-local operator with variable exponents and concave–convex nonlinearity (2018). arXiv:1810.12960v2 [math.AP]

  12. Brändle, C., Colorado, E., de Pablo, A., Sánchez, U.: A concave–convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. 143, 39–71 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Caffarelli, L.: Nonlocal equations, drifts and games. In: Nonlinear Partial Differential Equations. Abel Symposia, vol. 7, pp. 37–52. Springer, Berlin (2012)

    Google Scholar 

  14. Carvalho, M.L.M., da Silva, E.D., Goulart, C.: Quasilinear elliptic problems with concave–convex nonlinearities. Commun. Contemp. Math. 19, 1650050 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Chen, W., Deng, S.: The Nehari manifold for nonlocal elliptic operators involving concave–convex nonlinearities. Z. Angew. Math. Phys. 66, 1387–1400 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Choi, E.B., Kim, J.-M., Kim, Y.-H.: Infinitely many solutions for nonlinear elliptic equations of \(p(x)\)-Laplace type without the Ambrosetti and Rabinowitz condition. Proc. R. Soc. Edinb. 148, 1–31 (2018)

    Google Scholar 

  17. Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Dai, G., Hao, R.: Existence of solutions for a \(p(x)\)-Kirchhoff-type equation. J. Math. Anal. Appl. 359, 275–284 (2009)

    MathSciNet  MATH  Google Scholar 

  19. de Figueiredo, D.G., Gossez, J.P., Ubilla, P.: Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 199, 452–467 (2003)

    MathSciNet  MATH  Google Scholar 

  20. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Diening, L., Harjulehto, P., Hästö, P., Ru̇žička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)

    MATH  Google Scholar 

  22. Fabian, M., Habala, P., Hajék, P., Montesinos, V., Zizler, V.: Banach Space Theory: The Basis for Linear and Nonlinear Analysis. Springer, New York (2011)

    MATH  Google Scholar 

  23. Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m,p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Fan, X.L., Zhang, Q.H.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Ho, K., Kim, Y.-H.: A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional \(p(\cdot )\)-Laplacian. Nonlinear Anal. 188, 179–201 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Ho, K., Kim, Y.-H.: The concentration-compactness principles for \(W^{s,p(\cdot ,\cdot )}(\mathbb{R}^{N})\) and application. Adv. Nonlinear Anal. 10, 816–848 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Ho, K., Sim, I.: Existence and multiplicity of solutions for degenerate \(p(x)\)-Laplace equations involving concave–convex type nonlinearities with two parameters. Taiwan. J. Math. 19, 1469–1493 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landsman–Lazer type problem set on \(\mathbb{R}^{N}\). Proc. R. Soc. Edinb. A 129, 787–809 (1999)

    MATH  Google Scholar 

  30. Juárez Hurtado, E., Miyagaki, O.H., Rodrigues, R.S.: Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti–Rabinowitz type conditions. J. Dyn. Differ. Equ. 30, 405–432 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Kaufmann, U., Rossi, J.D., Vidal, R.: Fractional Sobolev spaces with variable exponents and fractional \(p(x)\)-Laplacians. Electron. J. Qual. Theory Differ. Equ. 2017, 76 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Kim, I.H., Kim, Y.-H.: Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents. Manuscr. Math. 147, 169–191 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Kim, I.H., Kim, Y.-H., Park, K.: Existence and multiplicity of solutions for Schrödinger–Kirchhoff type problems involving the fractional \(p(\cdot )\)-Laplacian in \(\mathbb{R}^{N}\). Bound. Value Probl. 2020, 121 (2020)

    Google Scholar 

  34. Kim, J.-M., Kim, Y.-H., Lee, J.: Existence of weak solutions to a class of Schrödinger type equations involving the fractional p-Laplacian in \(\mathbb{R}^{N}\). J. Korean Math. Soc. 56, 1441–1461 (2019)

    MathSciNet  Google Scholar 

  35. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). Czechoslov. Math. J. 41, 592–618 (1991)

    MATH  Google Scholar 

  36. Laskin, N.: Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 268, 298–305 (2000)

    MathSciNet  MATH  Google Scholar 

  37. Lee, J., Kim, J.-M., Kim, Y.-H.: Existence and multiplicity of solutions for Kirchhoff–Schrödinger type equations involving \(p(x)\)-Laplacian on the entire space \(\mathbb{R}^{N}\). Nonlinear Anal., Real World Appl. 45, 620–649 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Lee, J.I., Kim, J.-M., Kim, Y.-H., Lee, J.: Multiplicity of weak solutions to non-local elliptic equations involving the fractional \(p(x)\)-Laplacian. J. Math. Phys. 61, 011505 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Li, G., Yang, C.: The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti–Rabinowitz condition. Nonlinear Anal. 72, 4602–4613 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Lin, X., Tang, X.H.: Existence of infinitely many solutions for p-Laplacian equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 92, 72–81 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Liu, D.C.: On a \(p(x)\)-Kirchhoff-type equation via fountain theorem and dual fountain theorem. Nonlinear Anal. 72, 302–308 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Liu, S.B.: On ground states of superlinear p-Laplacian equations in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 361, 48–58 (2010)

    MathSciNet  Google Scholar 

  43. Mihăilescu, M., Rădulescu, V.: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc. Ser. A 462, 2625–2641 (2006)

    MathSciNet  MATH  Google Scholar 

  44. Oanh, B.T.K., Phuong, D.N.: On multiplicity solutions for a non-local fractional p-Laplace equation. Complex Var. Elliptic Equ. 65, 801–822 (2020)

    MathSciNet  MATH  Google Scholar 

  45. Papageorgiou, N.S., Scapellato, A.: Concave–convex problems for the Robin p-Laplacian plus an indefinite potential. Mathematics 8(3), 421 (2020)

    Google Scholar 

  46. Papageorgiou, N.S., Scapellato, A.: Nonlinear resonant problems with an indefinite potential and concave boundary condition. Electron. J. Qual. Theory Differ. Equ. 2020, 51 (2020)

    MathSciNet  MATH  Google Scholar 

  47. Papageorgiou, N.S., Scapellato, A.: Multiple solutions for Robin \((p,q)\)-equations plus an indefinite potential and a reaction concave near the origin. Anal. Math. Phys. 11, 78 (2021)

    MathSciNet  MATH  Google Scholar 

  48. Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in \(\mathbb{R}^{N}\). Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)

    MATH  Google Scholar 

  49. Ru̇žička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)

    Google Scholar 

  50. Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)

    MathSciNet  MATH  Google Scholar 

  51. Teng, K.: Multiple solutions for a class of fractional Schrödinger equations in \(\mathbb{R}^{N}\). Nonlinear Anal., Real World Appl. 21, 76–86 (2015)

    MathSciNet  MATH  Google Scholar 

  52. Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)

    MATH  Google Scholar 

  53. Wu, T.-F.: On semilinear elliptic equations involving concave–convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318, 253–270 (2006)

    MathSciNet  MATH  Google Scholar 

  54. Wu, T.-F.: Multiple positive solutions for a class of concave–convex elliptic problems in \(\mathbb{R}^{N}\) involving sign-changing weight. J. Funct. Anal. 258, 99–131 (2010)

    MathSciNet  MATH  Google Scholar 

  55. Xiang, M., Zhang, B., Ferrara, M.: Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities. Proc. R. Soc. A 471, 20150034 (2015)

    MathSciNet  MATH  Google Scholar 

  56. Zhong, C.-K.: A generalization of Ekeland’s variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 29, 1421–1431 (1997)

    MathSciNet  MATH  Google Scholar 

  57. Zhou, Y., Wang, J., Zhang, L.: Basic Theory of Fractional Differential Equations, 2nd edn. World Scientific, Singapore (2017)

    MATH  Google Scholar 

  58. Zuo, J., An, T., Fiscella, A.: A critical Kirchhoff-type problem driven by a \(p(\cdot )\)-fractional Laplace operator with variable \(s(\cdot )\)-order. Math. Methods Appl. Sci. 44, 1071–1085 (2021)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

A. Scapellato is a member of the INdAM (Istituto Nazionale di Alta Matematica “Francesco Severi”) research group GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni).

Funding

J. Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1C1C1008713). J.-M. Kim’s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521). Y.-H. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1F1A1057775).

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Lee, J., Kim, JM., Kim, YH. et al. On multiple solutions to a nonlocal fractional \(p(\cdot )\)-Laplacian problem with concave–convex nonlinearities. Adv Cont Discr Mod 2022, 14 (2022). https://doi.org/10.1186/s13662-022-03689-6

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MSC

  • 35B38
  • 35D30
  • 35J20
  • 35J60
  • 35J66

Keywords

  • Fractional \(p(x)\)-Laplacian
  • Variable exponent Lebesgue–Sobolev spaces
  • Weak solution
  • Fountain theorem
  • Dual fountain theorem