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Multiplicity of solutions for a class of fractional \(p(x,\cdot )\)-Kirchhoff-type problems without the Ambrosetti–Rabinowitz condition

Abstract

We are interested in the existence of solutions for the following fractional \(p(x,\cdot )\)-Kirchhoff-type problem:

$$ \textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases} $$

where \(\Omega \subset \mathbb{R}^{N}\), \(N\geq 2\) is a bounded smooth domain, \(s\in (0,1)\), \(p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )\), \((-\Delta )^{s}_{p(x,\cdot )}\) denotes the \(p(x,\cdot )\)-fractional Laplace operator, \(M: [0,\infty ) \to [0, \infty )\), and \(f: \Omega \times \mathbb{R} \to \mathbb{R}\) are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo–Benci–Fortunato (Nonlinear Anal. 7(9):981–1012, 1983), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti–Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

Introduction

Let Ω be a smooth bounded domain in \(\mathbb{R}^{N}\), \(N\geq 2\). Let us consider the following fractional \(p(x,\cdot )\)-Kirchhoff-type problem:

$$\begin{aligned} \textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(0< s<1\), \(p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )\) is a continuous function with \(sp(x,y)< N\) for all \((x,y) \in \overline{\Omega }\times \overline{\Omega }\), and M, f are continuous functions satisfying certain growth conditions to be stated later on.

The fractional \(p(x,\cdot )\)-Laplacian operator \((-\Delta )^{s}_{p(x,\cdot )}\) is, up to normalization factors by the Riesz potential, defined as follows: for each \(x\in \Omega \),

$$ (-\Delta )^{s}_{p(x,\cdot )} \varphi (x) = {\mathrm{{p.v.}}} \int _{\Omega } \frac{ \vert \varphi (x) - \varphi (y) \vert ^{p(x,y)-2} (\varphi (x) - \varphi (y))}{ \vert x-y \vert ^{N+sp(x,y)}} \,dy, $$
(1.2)

along any \(\varphi \in C^{\infty }_{0}(\Omega )\), where p.v. is the commonly used abbreviation for the principal value.

Throughout this paper, we shall assume that \(M: \mathbb{R}_{0}^{+} := [0, +\infty ) \to \mathbb{R}_{0}^{+}\) is a continuous function satisfying the following conditions:

\((M_{1})\): there exist \(\tau _{0} > 0\) and \(\gamma \in (1,{(p^{*}_{s})^{-}}/{p^{+}} )\) such that

$$ tM(t)\leq \gamma \widehat{M}(t), \quad \text{for all } t\geq \tau _{0}, $$

where

$$ \widehat{M}(t)= \int _{0}^{t}M(\tau ) \,d\tau $$

and \(p^{+}\) and \(p^{-}\) will be defined in Sect. 2;

\((M_{2})\): for every \(\tau >0\) there exists \(\kappa =\kappa (\tau )>0\) such that

$$ M(t)\geq \kappa , \quad \text{for all } t\geq \tau . $$

Obviously, the conditions \((M_{1})\) and \((M_{2})\) are fulfilled for the model case:

$$ M(t) = a + b \theta t^{\theta -1}, \quad \text{where } a, b \geq 0 \text{ and } a + b > 0. $$
(1.3)

It is worth pointing out that condition \((M_{2})\) was originally used to establish multiplicity of solutions for a class of higher order \(p(x)\)-Kirchhoff equations [11].

In recent years, a lot of attention has been given to problems involving fractional and nonlocal operators. This type of operators arises in a natural way in many different applications, e.g., image processing, quantum mechanics, elastic mechanics, electrorheological fluids (see [8, 15, 16, 34] and the references therein).

In their pioneering paper, Bahrouni and Rădulescu [6] studied qualitative properties of the fractional Sobolev space \(W^{s,q(x),p(x,y)}(\Omega )\), where Ω is a smooth bounded domain. Their results have been applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

Recently, by means of approximation and energy methods, Zhang and Zhang [38] have established the existence and uniqueness of nonnegative renormalized solutions for such problems. When \(s=1\), the operator degrades to integer order. It has been extensively studied in the literature; see for example [9, 10, 18, 19, 21] and the references therein. In particular, when \(p(x,\cdot )\) is a constant, this operator is reduced to the classical fractional p-Laplacian operator.

For studies concerning this operator, we refer to [31, 32, 3941]. We emphasize that, unless the functions \(p(x,\cdot)\) and \(q(x)\) are constants, the space \(W^{s,q(x),p(x,y)}(\Omega )\) does not coincide with the Sobolev space \(W^{s,p(x)}(\Omega )\) when s is a natural number; see [16, 23]. However, because of various applications in physics and in mathematical finance, the study of nonlocal problems in such spaces is still very interesting.

On the other hand, a lot of interest has in recent years been devoted to the study of Kirchhoff-type problems. More precisely, in 1883 Kirchhoff [24] established a model given by the following equation:

$$ \rho \frac{\partial ^{2}u}{\partial t^{2}} - \biggl( \frac{p_{0}}{\lambda }+\frac{E}{2L} \int _{0}^{L} \biggl\vert \frac{\partial u}{\partial x} \biggr\vert ^{2}\,dx \biggr) \frac{\partial ^{2}u}{\partial x^{2}}=0, $$
(1.4)

a generalization of the well-known D’Alembert wave equation for free vibrations of elastic strings, where ρ, \(p_{0}\), λ, E, L are constants which represent some physical meanings, respectively.

In the study of problem (1.1), the following Ambrosetti–Rabinowitz condition given in [3] has been widely used:

(AR): There exists a constant \(\mu >p^{+}\) such that

$$ t f(x,t) \geq \mu F(x,t) > 0, \quad \text{where } F(x,t)= \int ^{t}_{0}f(x,s) \,ds. $$

Clearly, if the \((AR)\) condition holds, then

$$\begin{aligned} F(x,t)\geq c_{1} \vert t \vert ^{\mu }-c_{2}, \quad \text{for all } (x,t)\in \Omega \times \mathbb{R}, \end{aligned}$$
(1.5)

where \(c_{1}\), \(c_{2}\) are two positive constants.

It is well known that \((AR)\) condition is very important for ensuring the boundedness of the Palais–Smale sequence. When the nonlinear term f satisfies the \((AR)\) condition, many results have been obtained by using the critical point theory and variational methods; see for example [1, 2, 4, 5, 1214, 17, 20, 28, 29, 36, 37]. In particular, Ali et al. [1] and Azroul et al. [5] have established the existence of nontrivial weak solutions for a class of fractional \(p(x,\cdot )\)-Kirchhoff-type problems by using the mountain pass theorem of Ambrosetti and Rabinowitz, direct variational approach, and Ekeland’s variational principle.

Since the \((AR)\) condition implies condition (1.5), one cannot deal with problem (1.1) by using the mountain pass theorem directly if \(f(x, t)\) is \(p^{+}\)-asymptotically linear at ∞, i.e.

$$ \lim_{ \vert t \vert \to \infty } \frac{f(x, t)}{ \vert t \vert ^{p^{+}-1 }}= l, \quad \text{uniformly in } x \in \Omega , $$
(1.6)

where l is a constant. For this reason, in recent years some authors have studied problem (1.1) by trying to omit the condition \({(AR)}\); see for example [18, 22, 27].

Not having the \((AR)\) condition brings great difficulties, so it is natural to consider if this kind of fractional problems have corresponding results even if the nonlinearity does not satisfy the \((AR)\) condition. In fact, in the absence of Kirchhoff’s interference, Lee et al. [26] have obtained infinitely many solutions to a fractional \(p(x)\)-Laplacian equation without assuming the \((AR)\) condition, by using the fountain theorem and the dual fountain theorem.

Inspired by the above work, we consider in this paper the fractional \(p(x,\cdot )\)-Kirchhoff-type problem without the \((AR)\) condition. Our situation is different from [1, 5] since our Kirchhoff function M belongs to a larger class of functions, whereas the nonlinear term f is \(p^{+}\)-asymptotically linear at ∞.

More precisely, let us assume that f satisfies the following global conditions:

\((F_{1})\)::

\(f: \Omega \times \mathbb{R}\to \mathbb{R}\) is continuous with \(F(x,t) \geq 0\), for all \((x,t) \in \Omega \times \mathbb{R}\), where

$$ F(x,t)= \int ^{t}_{0}f(x,s) \,ds; $$
\((F_{2})\)::

there exist a function \(\alpha \in C(\overline{\Omega })\), \(p^{+}<\alpha ^{-} \leq \alpha (x)<p_{s}^{*}(x)\), for all \(x\in \Omega \), and a number \(\Lambda _{0}>0\) such that, for each \(\lambda \in (0, \Lambda _{0})\), \(\epsilon >0\), there exists \(C_{\epsilon }>0\) such that

$$ f(x,t)\leq (\lambda +\epsilon ) \vert t \vert ^{\overline{p}(x)-1}+ C_{\epsilon } \vert t \vert ^{\alpha (x)-1}, \quad \text{for all } (x,t) \in \Omega \times \mathbb{R}; $$
\((F_{3})\)::

the following is uniformly satisfied on Ω̅:

$$ \lim_{ \vert t \vert \to \infty } \frac{F(x, t)}{ \vert t \vert ^{p^{+}\gamma } }= \infty ; $$
\((F_{4})\)::

there exist constants \(\mu >p^{+}\gamma \) and \(\varpi _{0}>0\) such that

$$ F(x,t) \leq \frac{1}{\mu } f(x,t) t + \varpi _{0} \vert t \vert ^{p^{-}}, \quad \text{for all } (x,t)\in \Omega \times \mathbb{R}; $$

where γ is given by \((M_{1})\);

\((F_{5})\)::

the following holds:

$$ f(x,-t)=-f(x,t), \quad \text{for all } (x,t)\in \Omega \times \mathbb{R}. $$
A simple computation proves that the function

$$\begin{aligned} f(x,t)= \vert t \vert ^{p^{+}\gamma -2}t \ln ^{\alpha (x)} \bigl(1+ \vert t \vert \bigr), \quad \text{where } \alpha (x)> 1, \end{aligned}$$
(1.7)

does not satisfy the \({(AR)}\) condition. However, it is easy to see that \(f(x,t)\) in (1.7) satisfies conditions \((F_{1})\)\((F_{5})\).

We can now state the definition of (weak) solutions for problem (1.1) (see Sect. 2 for details):

Definition 1.1

A function \(u\in E_{0}=W_{0}^{s,q(x),p(x,y)}(\Omega )\) is called a (weak) solution of problem (1.1), if for every \(w\in E_{0}\) it satisfies the following:

$$ \begin{aligned} &M \bigl(\sigma _{p(x,y)}(u) \bigr) \int _{\Omega \times \Omega } { \frac{ \vert u(x)-u(y) \vert ^{p(x,y)-2} (u(x)-u(y))(w(x)-w(y))}{ \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy \\ &\quad {}- \int _{\Omega }f(x,u) w \,dx=0, \end{aligned} $$

where

$$ \sigma _{p(x,y)}(u)= \int _{\Omega \times \Omega }\frac{1}{p(x,y)} \frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+s{p(x,y)}}} \,dx \,dy. $$

The main result of our paper is the following theorem.

Theorem 1.1

Let \(q(x)\), \(p(x,y)\)be continuous variable functions such that \(sp(x,y)< N\), \(p(x,y) = p(y,x)\)for all \((x,y)\in \overline{\Omega }\times \overline{\Omega }\)and \(q(x) \geq p(x,x)\)for all \(x \in \overline{\Omega }\). Assume that \(f:\Omega \times \mathbb{R}\to \mathbb{R}\)satisfies conditions \((F_{1})\)\((F_{5})\)and that \(M: \mathbb{R}_{0}^{+}\to \mathbb{R}_{0}^{+}\)is a continuous function satisfying conditions \((M_{1})\)and \((M_{2})\). Then there exists \(\Lambda >0\)such that, for each \(\lambda \in (0,\Lambda )\), problem (1.1) has a sequence \(\{u_{n}\}_{n}\)of nontrivial solutions.

The paper is organized as follows. In Sect. 2, we shall introduce the necessary properties of variable exponent Lebesgue spaces and fractional Sobolev spaces with variable exponent. In Sect. 3, we shall verify the Cerami compactness condition. Finally, in Sect. 4, we shall prove Theorem 1.1 by means of a version of the mountain pass theorem.

Fractional Sobolev spaces with variable exponent

For a smooth bounded domain Ω in \(\mathbb{R}^{N}\), we consider a continuous function \(p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )\). We assume that p is symmetric, that is,

$$ p(x,y)=p(y,x), \quad \text{for all } (x,y)\in \overline{\Omega }\times \overline{\Omega } $$

and

$$ 1< p^{-}:= \min _{(x,y)\in \overline{\Omega }\times \overline{ \Omega }}p(x,y) \leq p(x,y)\leq p^{+} = \max _{(x,y)\in \overline{\Omega }\times \overline{\Omega }}p(x,y)< \infty . $$

We also introduce a continuous function \(q: \overline{\Omega } \rightarrow \mathbb{R}\) such that

$$ 1< q^{-} := \min _{x\in \overline{\Omega }} q(x) \leq q(x) \leq q^{+} :=\max _{x\in \overline{\Omega }} q(x)< \infty . $$

We first give some basic properties of variable exponent Lebesgue spaces. Set

$$ C_{+}(\overline{\Omega })= \bigl\{ r\in C(\overline{\Omega }):1< r(x) \text{ for all }x\in \overline{\Omega } \bigr\} . $$

Given \(r\in C_{+}(\overline{\Omega }) \), we define the variable exponent Lebesgue space as

$$ L^{r(x)} (\Omega )= \biggl\{ u: \Omega \rightarrow \mathbb{R} \text{ is measurable: } \int _{\Omega } \bigl\vert u(x) \bigr\vert ^{r(x)} \,dx< \infty \biggr\} , $$

and this space is endowed with the Luxemburg norm,

$$ \vert u \vert _{r(x)}=\inf \biggl\{ \mu >0: \int _{\Omega } \biggl\vert \frac{u(x)}{\mu } \biggr\vert ^{r(x)}\,dx\leq 1 \biggr\} . $$

Then \((L^{r(x)}(\Omega ), |\cdot |_{r(x)})\) is a separable reflexive Banach space; see [25, Theorem 2.5 and Corollaries 2.7 and 2.12].

Let \(\widetilde{r}\in C_{+}(\overline{\Omega })\) be the conjugate exponent of q, that is,

$$ \frac{1}{r(x)}+\frac{1}{\widetilde{r}(x)}=1\quad \text{for all }x\in \overline{ \Omega }. $$

We shall need the following Hölder inequality, whose proof can be found in [25, Theorem 2.1]. Assume that \(v\in L^{r(x)}(\Omega )\) and \(u\in L^{\widetilde{r}(x)}(\Omega )\). Then

$$ \biggl\vert \int _{\Omega }uv \,dx \biggr\vert \leq \biggl( \frac{1}{r^{-}}+ \frac{1}{\widetilde{r}^{-}} \biggr) \vert u \vert _{r(x)} \vert v \vert _{\widetilde{r}(\cdot )}\leq 2 \vert u \vert _{r(x)} \vert v \vert _{ \widetilde{r}(x)}. $$

A modular of the \(L^{r(x)}(\Omega )\) space is defined by

$$ \varrho _{r(x)}:L^{r(x)}(\Omega )\rightarrow \mathbb{R}, \qquad u \mapsto \varrho _{r(x)}(u)= \int _{\Omega } \bigl\vert u(x) \bigr\vert ^{r(x)} \,dx. $$

Assume that \(u\in L^{r(x)}(\Omega )\) and \(\{u_{n}\}\subset L^{r(x)}(\Omega )\). Then the following assertions hold (see [16]):

$$\begin{aligned} (1) \quad & \vert u \vert _{r(x)}< 1 \quad (\text{resp.,}=1,>1) \quad \Leftrightarrow\quad \varrho _{r(x)}(u)< 1 \quad (\text{resp.,}=1,>1), \\ (2) \quad & \vert u \vert _{r(x)}< 1\quad \Rightarrow\quad \vert u \vert _{r(x)}^{r^{+}}\leq \varrho _{r(x)}(u)\leq \vert u \vert _{r(x)}^{r^{-}}, \\ (3) \quad & \vert u \vert _{r(x)}>1\quad \Rightarrow\quad \vert u \vert _{r(x)}^{r^{-}}\leq \varrho _{r(x)}(u)\leq \vert u \vert _{r(x)}^{r^{+}}, \\ (4) \quad &\lim_{n\rightarrow \infty } \vert u_{n} \vert _{r(x)}=0 \quad (\text{resp.,} = \infty )\quad \Leftrightarrow \quad \lim _{n\rightarrow \infty }\varrho _{r(x)}(u_{n})=0 \quad ( \text{resp.,} = \infty ), \\ (5) \quad &\lim_{n\rightarrow \infty } \vert u_{n}-u \vert _{r(x)}=0 \quad \Leftrightarrow\quad \lim_{n\rightarrow \infty }\varrho _{r(x)}(u_{n}-u)=0. \end{aligned}$$

Given \(s\in (0,1)\) and the functions \(p(x,y)\), \(q(x)\) as we mentioned above, the fractional Sobolev space with variable exponents via the Gagliardo approach \(E=W^{s,q(x),p(x,y)}(\Omega )\) is defined as follows:

$$ E= {\biggl\{ }u\in L^{q(x)}(\Omega ): \int _{\Omega \times \Omega } \frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{\mu ^{p(x,y)} \vert x-y \vert ^{N+sp(x,y)}} \,dx \,dy< \infty , \text{ for some } \mu >0 {\biggr\} }. $$

Let

$$ [u]_{s,p(x,y)}= \inf {\biggl\{ }\mu >0: \int _{\Omega \times \Omega } \frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{\mu ^{p(x,y)} \vert x-y \vert ^{N+sp(x,y)}} \,dx \,dy< 1 {\biggr\} }, $$

be the variable exponent Gagliardo seminorm and define

$$ \Vert u \Vert _{E}= [u]_{s,p(x,y)}+ \vert u \vert _{q(x)}. $$

Then E equipped with the norm \(\|\cdot \|_{E}\) becomes a Banach space.

Proposition 2.1

The following properties hold:

  1. (1)

    If \(1\leq [u]_{s,p(x,y)}<\infty \), then

    $$ \bigl([u]_{s,p(x,y)}\bigr)^{p_{-}}\leq \int _{\Omega \times \Omega } \frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+sp(x,y)}} \,dx \,dy\leq \bigl([u]_{s,p(x,y)} \bigr)^{p^{+}}. $$
  2. (2)

    If \([u]_{s,p(x,y)}\leq 1\), then

    $$ \bigl([u]_{s,p(x,y)}\bigr)^{p^{+}}\leq \int _{\Omega \times \Omega } \frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+sp(x,y)}} \,dx \,dy\leq \bigl([u]_{s,p(x,y)} \bigr)^{p_{-}}. $$

Given \(u\in W^{s,q(x),p(x,y)}(\Omega )\), we set

$$ \rho (u)= \int _{\Omega \times \Omega } \frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+sp(x,y)}} \,dx \,dy + \int _{\Omega } \vert u \vert ^{q(x)} \,dx $$

and

$$ \Vert u \Vert _{\rho }= \inf {\biggl\{ } \mu >0: \rho \biggl(\frac{u}{\mu } \biggr) \leq 1 {\biggr\} }. $$

It is well known that \(\|\cdot \|_{\rho }\) is a norm which is equivalent to the norm \(\|\cdot \|_{W^{s,q(x),p(x,y)}(\Omega )}\). By Lemma 2.2 in [38], \((W^{s,q(x),p(x,y)}(\Omega ), \|\cdot \|_{\rho })\) is uniformly convex and \(W^{s,q(x),p(x,y)}(\Omega )\) is a reflexive Banach space.

We denote our workspace \(E_{0}= W_{0}^{s,q(x),p(x,y)}(\Omega )\), the closure of \(C_{0}^{\infty }(\Omega )\) in E. Then \(E_{0}\) is a reflexive Banach space with the norm

$$ \Vert \cdot \Vert _{E_{0}}=[u]_{s,p(x,y)}. $$

A thorough variational analysis of the problems with variable exponents has been developed in the monograph by Rădulescu and Repovš [33]. The following result provides a compact embedding into variable exponent Lebesgue spaces.

Theorem 2.1

(see [38])

Let \(\Omega \subset \mathbb{R}^{n}\)be a smooth bounded domain and \(s\in (0,1)\). Let \(q(x)\), \(p(x,y)\)be continuous variable exponents such that

$$ sp(x,y)< N, \quad \textit{for } (x,y)\in \overline{\Omega }\times \overline{\Omega } \quad \textit{and} \quad q(x) \geq p(x,x), \quad \textit{for all } x \in \overline{ \Omega }. $$

Assume that \(\tau : \overline{\Omega }\longrightarrow (1,\infty )\)is a continuous function such that

$$ p^{*}(x)= \frac{Np(x,x)}{N-sp(x,x)} > \tau (x) \geq \tau ^{-}>1, \quad \textit{for all } x\in \overline{\Omega }. $$

Then there exists a constant \(C=C(N,s,p,q,r,\Omega )\)such that, for every \(u\in W^{s,q(x),p(x,y)}\),

$$ \vert u \vert _{\tau (x)} \leq C \Vert u \Vert _{E}. $$
(2.1)

That is, the space \(W^{s,q(x),p(x,y)}(\Omega )\)is continuously embeddable in \(L^{\tau (x)}(\Omega )\). Moreover, this embedding is compact. In addition, if \(u\in W_{0}^{s,q(x),p(x,y)}\), the following inequality holds:

$$ \vert u \vert _{\tau (x)}\leq C \Vert u \Vert _{E_{0}}. $$

Theorem 2.2

(see [6])

For all \(u, v\in E_{0}\), we consider the operator \(I:E_{0}\to {E^{*}_{0}}\)such that

$$ \bigl\langle I(u),v\bigr\rangle = \int _{\Omega \times \Omega } \frac{ \vert u(x)-u(y) \vert ^{p(x,y)-2} (u(x)-u(y))(v(x)-v(y))}{ \vert x-y \vert ^{N+sp(x,y)}} \,dx \,dy. $$

Then the following properties hold:

  1. (1)

    I is a bounded and strictly monotone operator.

  2. (2)

    I is a mapping of type \((S_{+})\), that is,

    $$ \textit{if } u_{n}\rightharpoonup u \in E_{0} \quad \textit{and} \quad \limsup_{n\rightarrow \infty } I(u_{n}) (u_{n}-u)\leq 0, \quad \textit{then } u_{n}\rightarrow u \in E_{0}. $$
  3. (3)

    \(I:E_{0}\to {E^{*}_{0}}\)is a homeomorphism.

The cerami compactness condition

Let us consider the Euler–Lagrange functional associated to problem (1.1), defined by \(J_{\lambda }: E_{0} \to \mathbb{R}\)

$$\begin{aligned} J_{\lambda }(u) =&\widehat{M} \bigl(\sigma _{p(x,y)}(u) \bigr)- \int _{ \Omega }F(x,u) \,dx. \end{aligned}$$
(3.1)

Note that \(J_{\lambda }\) is a \(C^{1}(E_{0}, \mathbb{R})\) functional and

$$\begin{aligned} \bigl\langle J_{\lambda }'(u), {w}\bigr\rangle &=M \bigl(\sigma _{p(x,y)}(u) \bigr) \int _{\Omega \times \Omega } { \frac{ \vert u(x)-u(y) \vert ^{p(x,y)-2} (u(x)-u(y))(w(x)-w(y))}{ \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy \\ &\quad {}- \int _{\Omega }f(x,u) w \,dx \end{aligned}$$
(3.2)

for all \(w\in E_{0}\). Therefore critical points of \(J_{\lambda }\) are weak solutions of problem (1.1).

In order to prove our main result (Theorem 1.1), we recall the definition of the Cerami compactness condition [30].

Definition 3.1

We say that \(J_{\lambda }\) satisfies the Cerami compactness condition at the level \(c \in \mathbb{R}\) ((Ce)c condition for short), if every sequence \(\{u_{n}\}_{n}\subset E_{0}\), i.e., \(J_{\lambda }(u_{n}) \to c\) and

$$ \bigl\Vert J_{\lambda }'(u_{n}) \bigr\Vert _{{E_{0}}^{*}}\bigl(1+ \Vert u_{n} \Vert _{E_{0}}\bigr)\to 0, \quad \text{as } n\to \infty , $$

admits a strongly convergent subsequence in \(E_{0}\). If \(J_{\lambda }\) satisfies the (Ce)c condition for any \(c \in \mathbb{R}\) then we say that \(J_{\lambda }\) satisfies the Cerami compactness condition.

Claim 3.1

Under assumptions of Theorem 1.1, every \((Ce)_{c}\)sequence \(\{u_{n}\}_{n}\subset E_{0}\)of \(J_{\lambda }\)is bounded in \(E_{0}\).

Proof

Let \(\{u_{n}\}_{n}\) be a \((Ce)_{c}\)-sequence of \(J_{\lambda }\). Then

$$ J_{\lambda }(u_{n})\to c \quad \text{and} \quad \bigl\Vert J_{\lambda }'(u_{n}) \bigr\Vert _{{E_{0}}^{*}}\bigl(1+ \Vert u_{n} \Vert _{E_{0}}\bigr)\to 0. $$
(3.3)

First, we prove that the sequence \(\{u_{n}\}_{n}\) is bounded in \(E_{0}\). To this end, we argue by contradiction. So suppose that \(\|u_{n}\|_{E_{0}} \to \infty \), as \(n\to \infty \). We define the sequence \(\{v_{n}\}_{n}\) by

$$ v_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert _{E_{0}}}, \quad n \in \mathbb{N}. $$

It is clear that \(\{v_{n}\}_{n}\subset E_{0}\) and \(\|v_{n}\|_{E_{0}} = 1\) for all \(n\in \mathbb{N}\). Passing, if necessary, to a subsequence, we may assume that

$$\begin{aligned}& v_{n} \rightharpoonup v\quad \text{in } E_{0}, \\& v_{n} \to v \quad \text{in } L^{\tau (x)}(\Omega ), 1\leq \tau (x)< p^{*}(x), \\& v_{n}(x) \to v(x) \quad \text{a.e. on } \Omega . \end{aligned}$$
(3.4)

Let \(\Omega _{\natural }:=\{x \in \Omega : v(x)\neq 0\}\). If \(x \in \Omega _{\natural }\), then it follows from (3.4) that

$$ \lim_{n\to \infty }v_{n}(x)=\lim_{n\to \infty } \frac{u_{n}}{ \Vert u_{n} \Vert _{E_{0}}}=v(x)\neq 0. $$

This means that

$$ \bigl\vert u_{n}(x) \bigr\vert = \bigl\vert v_{n}(x) \bigr\vert \Vert u_{n} \Vert _{E_{0}}\to +\infty \quad \text{a.e. on }\Omega _{\natural }, \text{ as } n\to \infty . $$

Moreover, it follows by condition \((F_{3})\) and Fatou’s lemma that, for each \(x\in \Omega _{\natural }\),

$$\begin{aligned} +\infty =\lim_{n\to \infty } \int _{\Omega } \frac{ \vert F(x,u_{n}(x)) \vert }{ \vert u_{n}(x) \vert ^{p_{+}\gamma }} \frac{ \vert u_{n}(x) \vert ^{p_{+}\gamma }}{ \Vert u_{n}(x) \Vert ^{p_{+}\gamma }_{E_{0}}} \,dx=\lim _{n\to \infty } \int _{\Omega } \frac{ \vert F(x,u_{n}(x)) \vert \vert v_{n}(x) \vert ^{p_{+}\gamma }}{ \vert u_{n}(x) \vert ^{p_{+}\gamma }} \,dx. \end{aligned}$$
(3.5)

Condition \((M_{1})\) gives

$$ \widehat{M}(t) \leq \widehat{M}(1) t^{\gamma }, \quad \text{for all } t\geq 1. $$
(3.6)

Now, since \(\|u_{n}\|_{E_{0}}>1\), it follows by (3.1), (3.3) and (3.6) that

$$\begin{aligned} \int _{\Omega }F(x,u_{n}) \,dx \leq &\widehat{M} (\sigma _{p(x,y)u_{n}} )+C \\ \leq &\frac{\widehat{M}(1)}{(p^{-})^{\gamma }} \bigl(\sigma _{p(x,y)}(u_{n}) \bigr)^{\gamma }+C \\ \leq & \frac{\widehat{M}(1)}{(p^{-})^{\gamma }} \Vert u_{n} \Vert _{E_{0}}^{\gamma p^{+}}+C, \end{aligned}$$

for all \(n\in \mathbb{N}\). We can now conclude that

$$\begin{aligned} \lim_{n\to \infty } \int _{\Omega } \frac{F(x,u_{n})}{ \Vert u_{n} \Vert _{E_{0}}^{p_{+}\gamma }} \,dx \leq & \lim _{n\to \infty } \biggl( \frac{\widehat{M}(1)}{(p^{-})^{\gamma }}+ \frac{C}{ \Vert u_{n} \Vert _{E_{0}}^{\gamma p^{+}}} \biggr). \end{aligned}$$
(3.7)

From (3.5) and (3.7) we obtain

$$ +\infty \leq \frac{\widehat{M}(1)}{(p^{-})^{\gamma }}, $$

which is a contradiction. Therefore

$$ \vert \Omega _{\natural } \vert = 0 \quad \text{and} \quad v(x) = 0 \quad \text{a.e. on } \Omega . $$

It follows from \((M_{1})\), \((M_{2})\), \((F_{4})\) and \(v_{n}\rightarrow v=0\) in \(L^{p^{-}}(\Omega )\) that

$$\begin{aligned} & \frac{1}{ \Vert u_{n} \Vert _{E_{0}}^{p^{-}}} \biggl(J_{\lambda }(u_{n})- \frac{1}{\mu }J_{\lambda }^{\prime } (u_{n}) u_{n} \biggr) \\ &\quad \geq \frac{1}{ \Vert u_{n} \Vert _{E_{0}}^{p^{-}}} \biggl[\widehat{M} \bigl(\sigma _{p(x,y)}(u_{n}) \bigr)- \int _{\Omega }F(x,u_{n}) \,dx \\ &\qquad {}-\frac{1}{\mu } M \bigl(\sigma _{p(x,y)}(u_{n}) \bigr) \int _{\Omega \times \Omega } {\frac{ \vert u_{n}(x)-u_{n}(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy +\frac{1}{\mu } \int _{\Omega }f(x,u_{n}) u_{n} \,dx \biggr] \\ &\quad \geq \frac{1}{ \Vert u_{n} \Vert _{E_{0}}^{p^{-}}} \biggl[\frac{1}{\gamma } M \bigl( \sigma _{p(x,y)}(u_{n}) \bigr) \sigma _{p(x,y)}(u_{n}) \\ &\qquad {}- \frac{1}{\mu } M \bigl( \sigma _{p(x,y)}(u_{n}) \bigr) \int _{\Omega \times \Omega } {\frac{ \vert u_{n}(x)-u_{n}(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy-\varpi _{0} \int _{\Omega } \vert u_{n} \vert ^{p^{-}}\,dx \biggr] \\ &\quad \geq \frac{1}{ \Vert u_{n} \Vert _{E_{0}}^{p^{-}}} \biggl[ \biggl(\frac{1}{\gamma p^{+}}-\frac{1}{\mu } \biggr) M \bigl(\sigma _{p(x,y)}(u_{n}) \bigr) \int _{\Omega \times \Omega } {\frac{ \vert u_{n}(x)-u_{n}(y) \vert ^{p(x,y)}}{ \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy\\ &\qquad {} - \varpi _{0} \int _{\Omega } \vert u_{n} \vert ^{p^{-}} \,dx \biggr] \\ &\quad \geq \biggl(\frac{1}{\gamma p^{+}}-\frac{1}{\mu } \biggr) \kappa -\lambda \varpi _{0} \int _{\Omega } \vert v_{n} \vert ^{p^{-}} \,dx, \end{aligned}$$

which means that

$$ 0\geq \biggl(\frac{1}{\gamma p^{+}}-\frac{1}{\mu } \biggr) \kappa , \quad \text{as } n\rightarrow \infty . $$

This is a contradiction. As a consequence, we can conclude that Cerami sequence \(\{u_{n}\}_{n}\) is indeed bounded. This completes the proof of Claim 3.1. □

We now complete the verification of the Cerami compactness condition \((Ce)_{c}\) for \(J_{\lambda }\).

Claim 3.2

The functional \(J_{\lambda }\)satisfies condition \((Ce)_{c}\)in \(E_{0}\).

Proof

Let \(\{u_{n}\}_{n}\) be a \((Ce)_{c}\) sequence for \(J_{\lambda }\) in \(E_{0}\). Claim 3.1 asserts that \(\{u_{n}\}_{n}\) is bounded in \(E_{0}\). By Theorem 2.1, the embedding \(E_{0} \hookrightarrow L^{\tau (x)}(\Omega )\) is compact, where \(1\leq \tau (x)< p^{*}(x)\). Since \(E_{0}\) is a reflexive Banach space, passing, if necessary, to a subsequence, still denoted by \(\{u_{n}\}_{n}\), there exists \(u\in E_{0}\) such that

$$ u_{n}\rightharpoonup u\quad \text{in } E_{0}, \qquad u_{n} \to u \quad \text{in } L^{ \tau (x)}(\Omega ),\qquad u_{n}(x)\to u(x),\quad \text{a.e. on } \Omega . $$
(3.8)

By virtue of (3.3), we get

$$\begin{aligned} &\bigl\langle J_{\lambda }'(u_{n}), {u_{n}-u}\bigr\rangle \\ &\quad =M \bigl(\sigma _{p(x,y)}(u_{n}) \bigr) \\ &\qquad {}\times \int _{\Omega \times \Omega } { \frac{ \vert u_{n}(x)-u_{n}(y) \vert ^{p(x,y)-2} (u_{n}(x)-u_{n}(y)) ((u_{n}(x)-u(x))-(u_{n}(y)-u(y)))}{ \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy \\ &\qquad {} - \int _{\Omega }f(x,u_{n}) (u_{n}-u) \,dx \to 0. \end{aligned}$$
(3.9)

Now, by condition \((F_{2})\),

$$ \bigl\vert f(x,u_{n}) \bigr\vert \leq (\lambda + \epsilon ) \vert u_{n} \vert ^{\overline{p}(x)-1}+C_{\epsilon } \vert u_{n} \vert ^{\alpha (x)-1}. $$
(3.10)

It follows from (3.8), (3.10) and Proposition 2.1 that

$$\begin{aligned} &\biggl\vert \int _{\Omega }f(x,u_{n}) (u_{n}-u) \,dx \biggr\vert \\ &\quad \leq \int _{\Omega }(\lambda + \epsilon ) \vert u_{n} \vert ^{\overline{p}(x)-1} \vert u_{n}-u \vert \,dx + \int _{\Omega }C_{\epsilon } \vert u_{n} \vert ^{\alpha (x)-1} \vert u_{n}-u \vert \,dx \\ &\quad \leq (\lambda + \epsilon ) { \bigl\vert { \vert u_{n} \vert } ^{\overline{p}(x)-1} \bigr\vert }_{ \frac{\overline{p}(x)}{\overline{p}(x)-1}}{ \vert u_{n}-u \vert }_{\overline{p}(x)} +C_{\epsilon }{ \bigl\vert { \vert u_{n} \vert } ^{\alpha (x)-1} \bigr\vert }_{ \frac{\alpha (x)}{\alpha (x)-1}} \vert u_{n}-u \vert _{\alpha (x)} \\ &\quad \leq (\lambda + \epsilon )\max \bigl\{ \Vert u_{n} \Vert ^{p_{+}-1}_{E_{0}}, \Vert u_{n} \Vert ^{p_{-}-1}_{E_{0}} \bigr\} { \vert u_{n}-u \vert } _{p(x)} \\ &\qquad {} +C_{\epsilon }\max \bigl\{ \Vert u_{n} \Vert ^{\alpha _{+}-1}_{E_{0}}, \Vert u_{n} \Vert ^{\alpha _{-}-1}_{E_{0}} \bigr\} _{ \frac{\alpha (x)}{\alpha (x)-1}} \vert u_{n}-u \vert _{\alpha (x)} \\ &\quad \to 0, \quad \text{as } n\to \infty , \end{aligned}$$

which implies that

$$ \lim_{n\to \infty } \int _{\Omega }f(x,u_{n}) (u_{n}-u) \,dx=0. $$
(3.11)

Therefore we can infer from (3.9) and (3.11) that

$$ \begin{aligned} &M \bigl(\sigma _{p(x,y)}(u_{n}) \bigr) \\ &\qquad {}\times\int _{\Omega \times \Omega } { \frac{ \vert u_{n}(x)-u_{n}(y) \vert ^{p(x,y)-2} (u_{n}(x)-u_{n}(y)) ((u_{n}(x)-u(x))-(u_{n}(y)-u(y)))}{ \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy\\ &\quad \to 0. \end{aligned} $$

Since \(\{u_{n}\}_{n}\) is bounded in \(E_{0}\), using \((M_{2})\), we can conclude that the sequence of positive real numbers \(\{ M (\sigma _{p(x,y)}(u_{n}) ) \} \) is bounded from below by some positive number for n large enough. Invoking Theorem 2.2, we can deduce that \(u_{n} \to u\) strongly in \(E_{0}\). This completes the proof of Claim 3.2. □

Proof of Theorem 1.1

To prove Theorem 1.1, we shall use the following symmetric mountain pass theorem.

Theorem 4.1

(see [7, 35])

Let \(X=Y\oplus Z\)be an infinite-dimensional Banach space, where Y is finite-dimensional, and let \(I\in C^{1}(X,\mathbb{R})\). Suppose that:

  1. (1)

    I satisfies \((Ce)_{c}\)-condition, for all \(c>0\);

  2. (2)

    \(I(0)=0\), \(I(-u)=I(u)\), for all \(u\in X\);

  3. (3)

    there exist constants ρ, \(a>0\)such that \(I|_{\partial B\rho \cap Z} \geq a\);

  4. (4)

    for every finite-dimensional subspace \(\widetilde{X}\subset X\), there is \(R=R(\widetilde{X})>0\)such that \(I(u)\leq 0\)on \(\widetilde{X}\backslash B_{R}\).

Then I possesses an unbounded sequence of critical values.

Let us first verify that functional \(J_{\lambda }\) satisfies the mountain pass geometry.

Claim 4.1

Under the hypotheses of Theorem 1.1, there exists \(\Lambda >0\)such that, for each \(\lambda \in (0, \Lambda )\), we can choose \(\rho > 0\)and \(a > 0\)such that

$$ J_{\lambda }(u) \geq a > 0, \quad \textit{for all } u\in {E_{0}} \textit{ with } \Vert u \Vert =\rho . $$

Proof

Let \(\rho \in (0,1)\) and \(u\in E_{0}\) be such that \(\| u\|_{E_{0}} =\rho \). By assumption \((F_{2})\), for every \(\epsilon >0\), there exists \(C_{\epsilon }>0\) such that

$$ \bigl\vert F(x,t) \bigr\vert \leq \frac{(\lambda + \epsilon )}{\overline{p}(x)} \vert t \vert ^{ \overline{p}(x)}+\frac{C_{\epsilon }}{\alpha (x)} \vert t \vert ^{\alpha (x)}, \quad \text{for all } x\in \Omega , t\in \mathbb{R}. $$
(4.1)

Moreover, \((M_{2})\) gives

$$ \widehat{M}(t) \geq \widehat{M}(1) t^{\gamma }, \quad \text{for all } t\in [0, 1], $$
(4.2)

whereas \((M_{1})\) implies that \(\widehat{M}(1) > 0\). Thus, using (4.1), (4.2) and (2.1), we obtain, for all \(u\in {E_{0}}\), with \(\| u\|_{E_{0}} =\rho \),

$$\begin{aligned} J_{\lambda }(u) =&\widehat{M} \bigl(\sigma _{p(x,y)}(u) \bigr)- \int _{ \Omega }F(x,u) \,dx \\ \geq &\frac{\widehat{M}(1)}{(p^{+})^{\gamma }} \bigl(\sigma _{p(x,y)}(u) \bigr)^{\gamma }- \int _{\Omega }\frac{\lambda + \epsilon }{\overline{p}(x)} \vert u \vert ^{\overline{p}(x)} \,dx- \int _{\Omega }\frac{C_{\epsilon }}{\alpha (x)} \vert u \vert ^{\alpha (x)} \,dx \\ \geq &\frac{\widehat{M}(1)}{(p^{+})^{\gamma }}\min \bigl\{ \Vert u \Vert _{E_{0}}^{ \gamma p^{+}}, \Vert u \Vert _{E_{0}}^{\gamma p^{-}}\bigr\} - \frac{\epsilon + \lambda }{p^{-}}\max \bigl\{ \vert u \vert _{p(x)}^{ p^{+}}, \vert u \vert _{p(x)}^{ p^{-}}\bigr\} \\ &{}-\frac{C_{\epsilon }}{\alpha ^{-}}\max \bigl\{ \vert u \vert _{\alpha (x)}^{ \alpha ^{+}}, \vert u \vert _{\alpha (x)}^{\alpha ^{-}} \bigr\} \\ \geq &\frac{\widehat{M}(1)}{(p^{+})^{\gamma }} \Vert u \Vert _{E_{0}}^{\gamma p^{+}}-c_{1}( \epsilon + \lambda ) \Vert u \Vert _{E}^{ p^{-}}-c_{2} \Vert u \Vert _{E_{0}}^{\alpha ^{-}} \\ \geq & \rho ^{\gamma p^{+}} \biggl( \frac{\widehat{M}(1)}{(p^{+})^{\gamma }}-c_{1}( \epsilon + \lambda ) \rho ^{ p^{-}-\gamma p^{+}}-c_{2} \rho ^{\alpha ^{-}-\gamma p^{+}} \biggr), \end{aligned}$$
(4.3)

where \(\rho = \|u\|_{E_{0}}\). Since \(\epsilon >0\) is arbitrary, let us choose

$$ \epsilon = \frac{\widehat{M}(1)}{2c_{1}(p^{+})^{\gamma }} \rho ^{\gamma p^{+}-p^{-}} > 0. $$
(4.4)

Then, by (4.3) and (4.4), we obtain

$$\begin{aligned} J_{\lambda }(u) \geq \rho ^{\gamma p^{+}} \biggl( \frac{\widehat{M}(1)}{2(p^{+})^{\gamma }}-\lambda c_{1} \rho ^{ p^{-}-\gamma p^{+}}-c_{2} \rho ^{\alpha ^{-}-\gamma p^{+}} \biggr). \end{aligned}$$
(4.5)

Now, for each \(\lambda > 0\), we define a continuous function, \(g_{\lambda }: (0, \infty ) \to \mathbb{R}\),

$$ g_{\lambda }(s)=\lambda c_{1}s^{p^{-}-\gamma p^{+}}+c_{2}s^{\alpha ^{-}- \gamma p^{+}}. $$

Since \(1< p^{-}<\gamma p^{+}<\alpha ^{-}\), it follows that

$$ \lim_{s\to 0^{+}}g_{\lambda }(s) = \lim_{s\to +\infty }g_{\lambda }(s) =+ \infty . $$

Thus we can find the infimum of \(g_{\lambda }\). Note that equating

$$ g'_{\lambda }(s)=\lambda c_{1}\bigl(p^{-}- \gamma p^{+}\bigr)s^{p^{-}-\gamma p^{+}-1}+c_{2}\bigl( \alpha ^{-}-{\gamma p^{+}}\bigr)s^{\alpha ^{-}-{\gamma p^{+}}-1}=0, $$

we get

$$ s_{0}=s=\widetilde{C}\lambda ^{\frac{1}{\alpha ^{-}-p^{-}}}, $$

where

$$ \widetilde{C} := \biggl( \frac{ c_{1}(\gamma p^{+}-p^{-})}{c_{1}(\alpha ^{-}-\gamma p^{+})} \biggr)^{\frac{1}{\alpha ^{-}-p^{-}}} >0. $$

Clearly, \(s_{0} > 0\). It can also be checked that \(g''_{\lambda }(s_{0}) > 0\) and hence the infimum of \(g_{\lambda }(s)\) is achieved at \(s_{0}\).

Now, observing that

$$ g_{\lambda }(s_{0})= \bigl(c_{1}\widetilde{C}^{p^{-}-\gamma p^{+}}+c_{2} \widetilde{C}^{\alpha ^{-}-\gamma p^{+}} \bigr) \lambda ^{\frac{\alpha ^{-}-\gamma p^{+}}{\alpha ^{-}-p^{-}}} \rightarrow 0, \quad \text{as } \lambda \to 0^{+}, $$

we can infer from (4.5) that there exists \(0 < \Lambda < \Lambda _{0}\) (see \((F_{2})\)) such that all for all \(\lambda \in (0,\Lambda )\) we can choose ρ small enough and \(\alpha > 0\) such that

$$ J_{\lambda }(u) \geq a > 0, \quad \text{for all } u \in E_{0} \text{ with } \Vert u \Vert _{E_{0}}=\rho . $$

This completes the proof of Claim 4.1. □

Claim 4.2

Under the hypotheses of Theorem 1.1, for every finite-dimensional subspace \(W\subset E_{0}\)there exists \(R= R(W)>0\)such that

$$ J_{\lambda }(u)\leq 0, \quad \textit{for all } u\in W, \textit{ with } \Vert u \Vert _{E_{0}}\geq R. $$

Proof

In view of \((F_{3})\), we know that, for all \(A>0\), there exists \(C_{A}>0\) such that

$$ F(x,t)\geq A \vert t \vert ^{\gamma p^{+}}- C_{A}, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}. $$
(4.6)

Again, \((M_{2})\) gives

$$ \widehat{M}(t) \leq \widehat{M}(1) t^{\gamma }, \quad \text{for all } t\geq 1, $$
(4.7)

with \(\widehat{M}(1) > 0\) by \((M_{1})\). By (4.6) and (4.7) we have

$$\begin{aligned} J_{\lambda }(u)&=\widehat{M} \bigl(\sigma _{p(x,y)}(u) \bigr)- \int _{\Omega }F(x,u) \,dx \\ &\leq \frac{\widehat{M}(1)}{(p^{-})^{\gamma }} \bigl(\sigma _{p(x,y)}(u)\bigr)^{\gamma }- A \int _{\Omega } \vert u \vert ^{\gamma p^{+}} \,dx + C_{A} \vert \Omega \vert \\ &\leq \frac{\widehat{M}(1)}{(p^{-})^{\gamma }} \Vert u \Vert _{E_{0}}^{\gamma p^{+}} - A \int _{\Omega } \vert u \vert ^{\gamma p^{+}} \,dx + C_{A} \vert \Omega \vert . \end{aligned}$$

Consequently, since \(\| u\|_{E_{0}} > 1\), all norms on the finite-dimensional space W are equivalent, so there is \(C_{W} > 0\) such that

$$ \int _{\Omega } \vert u \vert ^{\gamma p^{+}}\,dx\geq C_{W} \Vert u \Vert _{E_{0}} ^{\gamma p^{+}}. $$

Let \(R= R(W)>0\). Then for all \(u\in W\) with \(\|u\|_{E_{0}}\geq R\) we obtain

$$ J_{\lambda }(u) \leq \Vert u \Vert _{E_{0}} ^{p^{+}\gamma } \biggl( \frac{\widehat{M}(1)}{(p^{-})^{\gamma }}- AC_{W} \biggr)+ C_{A} \vert \Omega \vert . $$
(4.8)

So choosing in inequality 4.8,

$$ A=\frac{2\widehat{M}(1)}{C_{W}(p^{-})^{\gamma }}, $$

we can conclude that

$$ J_{\lambda }(u)\leq 0, \quad \text{for all } u\in W \text{ with } \Vert u \Vert _{E_{0}}\geq R. $$

This completes the proof of Claim 4.2. □

Proof of Theorem 1.1

Obviously, \(J_{\lambda }(0)=0\) and by condition \((F_{5})\), \(J_{\lambda }\) is an even functional. Invoking Claims 3.1, 3.2, 4.1, and 4.2, and Theorem 4.1, we can now conclude that there indeed exists an unbounded sequence of solutions of problem (1.1). This completes the proof of Theorem 1.1. □

References

  1. 1.

    Ali, K.B., Hsini, M., Kefi, K., Chung, N.T.: On a nonlocal fractional \(p(\cdot,\cdot)\)-Laplacian problem with competing nonlinearities. Complex Anal. Oper. Theory 13(3), 1377–1399 (2019)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alves, C., Molica Bisci, G.: A compact embedding result for anisotropic Sobolev spaces associated to a strip-like domain and some applications. J. Math. Anal. Appl. (2019). Published online. https://doi.org/10.1016/j.jmaa.2019.123490

    Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ambrosio, V., D’Onofrio, L., Molica Bisci, G.: Perturbation methods for nonlocal Kirchhoff-type problems. Fract. Calc. Appl. Anal. 20, 829–853 (2017)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Azroul, E., Benkirane, A., Shimi, M., Srati, M.: On a class of fractional \(p(x)\)-Kirchhoff type problems. Appl. Anal. (2019). Published online. https://doi.org/10.1080/00036811.2019.1603372

    Article  MATH  Google Scholar 

  6. 6.

    Bahrouni, A., Rădulescu, V.D.: On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin. Dyn. Syst., Ser. S, 11(3), 379–389 (2018)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Bartolo, T., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7(9), 981–1012 (1983)

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    Google Scholar 

  9. 9.

    Chung, N.T.: Multiple solutions for a \(p(x)\)-Kirchhoff-type equation with sign-changing nonlinearities. Complex Var. Elliptic Equ. 58(12), 1637–1646 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chung, N.T., Ngo, Q.A.: Multiple solutions for a class of quasilinear elliptic equations of \(p(x)\)-Laplacian type with nonlinear boundary conditions. Proc. R. Soc. Edinb., Sect. A, Math. 140(2), 259–272 (2010)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Colasuonno, F., Pucci, P.: Multiplicity of solutions for \(p(x)\)-polyharmonic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Devillanova, G., Marano, C.G.: A free fractional viscous oscillator as a forced standard damped vibration. Fract. Calc. Appl. Anal. 19(2), 319–356 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Devillanova, G., Solimini, S.: Infinitely many positive solutions to some nonsymmetric scalar field equations: the planar case. Calc. Var. 52(3–4), 857–898 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Devillanova, G., Solimini, S.: Some remarks on profile decomposition theorems. Adv. Nonlinear Stud. 16(4), 795–805 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    MathSciNet  Article  Google Scholar 

  16. 16.

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

    Google Scholar 

  17. 17.

    Fiscella, A., Pucci, P., Zhang, B.L.: p-Fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities. Adv. Nonlinear Anal. 8, 1111–1131 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hamdani, M.K.: On a nonlocal asymmetric Kirchhoff problems. Asian-Eur. J. Math. 13(5, art. 2030001 (2020)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hamdani, M.K., Chung, N.T., Repovš, D.D.: New class of sixth-order nonhomogeneous \(p(x)\)-Kirchhoff problems with sign-changing weight functions. Submitted

  20. 20.

    Hamdani, M.K., Harrabi, A., Mtiri, F., Repovš, D.D.: Existence and multiplicity results for a new \(p(x)\)-Kirchhoff problem. Nonlinear Anal. 190, art. 111598 (2020)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hamdani, M.K., Repovš, D.D.: Existence of solutions for systems arising in electromagnetism. J. Math. Anal. Appl. 486(2), art. 123898 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Harrabi, A., Hamdani, M.K., Selmi, A.: Existence results of the zero mass polyharmonic system. Complex Var. Elliptic Equ. 65(10), 1613–1629 (2020). https://doi.org/10.1080/17476933.2019.1679794

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hästö, P., Ribeiro, A.M.: Characterization of the variable exponent Sobolev norm without derivatives. Commun. Contemp. Math. 19(3, art. 1650022 (2017)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    Google Scholar 

  25. 25.

    Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{1,p(x)}\). Czechoslov. Math. J. 41(5), 592–618 (1991)

    Article  Google Scholar 

  26. 26.

    Lee, J.I., Kim, J., Kim, Y., Lee, J.: Multiplicity of weak solutions to non-local elliptic equations involving the fractional \(p(x)\)-Laplacian. J. Math. Phys. 61(1), 011505 (2020)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Li, G., Rădulescu, V.D., Repovš, D.D., Zhang, Q.: Nonhomogeneous Dirichlet problems without the Ambrosetti–Rabinowitz condition. Topol. Methods Nonlinear Anal. 51(1), 55–77 (2018)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Mingqi, X., Molica Bisci, G., Tian, G., Zhang, B.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian. Nonlinearity 29, 357–374 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Molica Bisci, G., Radulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  30. 30.

    Papageorgiou, N.S., Rǎdulescu, V.D., Repovš, D.D.: Nonlinear Analysis—Theory and Methods. Springer, Cham (2019)

    Google Scholar 

  31. 31.

    Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \(\mathbb{R}^{N}\) involving non-local operators. Rev. Mat. Iberoam. 32, 1–22 (2016)

    MathSciNet  Article  Google Scholar 

  32. 32.

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

    MathSciNet  Article  Google Scholar 

  33. 33.

    Rădulescu, V.D., Repovš, D.D.: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis. Taylor & Francis, Boca Raton (2015)

    Google Scholar 

  34. 34.

    Ružička, M.: Electro-Rheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Math., vol. 1784. Springer, Berlin (2000)

    Google Scholar 

  35. 35.

    Tang, X.H.: Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 401, 407–415 (2013)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Xiang, M.Q., Zhang, B., Ferrara, M.: Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian. J. Math. Anal. Appl. 424, 1021–1041 (2015)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Xiang, M.Q., Zhang, B., Rădulescu, V.D.: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity 29, 3186–3205 (2016)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Zhang, C., Zhang, X.: Renormalized solutions for the fractional \(p(x)\)-Laplacian equation with \(L^{1}\) data. Nonlinear Anal. 190, art. 111610 (2020)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Zuo, J., An, T., Li, M.: Superlinear Kirchhoff-type problems of the fractional p-Laplacian without the (AR) condition. Bound. Value Probl. 2018, 180 (2018)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Zuo, J., An, T., Yang, L., Ren, X.: The Nehari manifold for a fractional p-Kirchhoff system involving sign-changing weight function and concave-convex nonlinearities. J. Funct. Spaces 2019, art. ID 7624373 (2019)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Zuo, J., An, T., Ye, G., Qiao, Z.: Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity. Electron. J. Qual. Theory Differ. Equ. 2019, 41 (2019)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their comments and suggestions.

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Funding

The first author was supported by the Tunisian Military Research Center for Science and Technology Laboratory. The second author was supported by the Fundamental Research Funds for Central Universities (2019B44914) and the National Key Research and Development Program of China (2018YFC1508100), the China Scholarship Council (201906710004), and the Jilin Province Undergraduate Training Program for Innovation and Entrepreneurship (201910204085). The third author was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (Grant N.101.02.2017.04). The fourth author was supported by the Slovenian Research Agency grants P1-0292, N1-0114, N1-0083, N1-0064, and J1-8131.

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Correspondence to D. D. Repovš.

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Hamdani, M.K., Zuo, J., Chung, N.T. et al. Multiplicity of solutions for a class of fractional \(p(x,\cdot )\)-Kirchhoff-type problems without the Ambrosetti–Rabinowitz condition. Bound Value Probl 2020, 150 (2020). https://doi.org/10.1186/s13661-020-01447-9

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MSC

  • 35R11
  • 35J20
  • 35J60

Keywords

  • Fractional \(p(x,\cdot )\)-Kirchhoff-type problems
  • \(p(x,\cdot )\)-fractional Laplace operator
  • Ambrosetti–Rabinowitz type conditions
  • Symmetric mountain pass theorem
  • Cerami compactness condition
  • Fractional Sobolev spaces with variable exponent
  • Multiplicity of solutions