Existence and Multiplicity of Solutions for a Bi-Non-Local Problem

Abstract

The aim of this paper is to investigate the existence and multiplicity of solutions for a bi-non-local problem. Precisely, we show that the above problem admits at least a non-trivial positive energy solution by using the mountain pass theorem. Furthermore, with the help of the fountain theorem, we obtain the existence of infinitely many positive energy solutions, assuming a symmetric condition for g. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters.

1. Introduction

Recently, Lorenzo and Hartley in [1] came up with the fractional variable-order derivatives that are used to describe different processes of nonlinear diffusion. Indeed, the variable order problem of non-local integro-differential operators can better reflect the temperature change of an object. Therefore, a large number of researchers have begun to pay attention to fractional variable-order spaces. See [2,3,4,5] and the references therein.

In this paper, we study the following variable $s(·)$-order fractional Kirchhoff type problem

$$\left\{\begin{array}{cc}M\left({\displaystyle {\int \int }_{{\mathbb{R}}^{2N}}\frac{1}{p(x,y)}\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy}\right){(-\Delta )}_{p(·)}^{s(·)}u\left(x\right)=\mu {\left|u\left(x\right)\right|}^{\overline{p}\left(x\right)-2}u\left(x\right)+g(x,u)\hfill & \mathrm{in}\Omega ,\hfill \\ u=0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(1)

where the domain $\Omega \subset {\mathbb{R}}^N$ is bounded and smooth with $N>p(x,y)s(x,y)$ for any $(x,y)\in \overline{\Omega }\times \overline{\Omega }$, where $\mu$ is a real parameter, $s(·):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (0,1)$ and $p(·):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (1,\infty )$, exponent $\overline{p}\left(x\right)=p(x,x)$ for $x\in \overline{\Omega }$. Here, ${(-\Delta )}_{p(·)}^{s(·)}$ is a $p(·)$-Laplace operator with fractional variable $s(·)$-order, which is given by

$${(-\Delta )}_{p(·)}^{s(·)}\phi \left(x\right)=P.V.{\int }_{{\mathbb{R}}^N}\frac{{|\phi \left(x\right)-\phi \left(y\right)|}^{p(x,y)-2}(\phi \left(x\right)-\phi \left(y\right))}{{|x-y|}^{N+p(x,y)s(x,y)}}dy,x\in {\mathbb{R}}^N,$$

(2)

along any $\phi \in C_0^{\infty }\left({\mathbb{R}}^N\right)$, where P.V. is the Cauchy principal value.

For the sake of convenience, we denote

$$\begin{array}{c}{s}^{-}=\underset{(x,y)\in {\mathbb{R}}^{2N}}{inf}s(x,y),s^{+}=\underset{(x,y)\in {\mathbb{R}}^{2N}}{sup}s(x,y),p^{-}=\underset{(x,y)\in {\mathbb{R}}^{2N}}{inf}p(x,y),p^{+}=\underset{(x,y)\in {\mathbb{R}}^{2N}}{sup}p(x,y),\\ p_s^{*}\left(x\right)=\frac{N\overline{p}\left(x\right)}{N-\overline{s}\left(x\right)\overline{p}\left(x\right)}\mathrm{with}\overline{p}\left(x\right)=p(x,x),\overline{s}\left(x\right)=s(x,x),{\overline{p}}^{-}=\underset{x\in {\mathbb{R}}^N}{inf}\overline{p}\left(x\right),{\overline{p}}^{+}=\underset{x\in {\mathbb{R}}^N}{sup}\overline{p}\left(x\right).\end{array}$$

The continuous function $M:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ fulfills the following conditions.

$\left(M_1\right)$There exist $h_2\ge h_1>0$ and $\beta >1$, with $p^{+}<\beta p^{-}$, such that

$$h_1{t}^{\beta -1}\le M\left(t\right)\le h_2{t}^{\beta -1}\mathit{for}\mathit{all}t\in {\mathbb{R}}^{+}.$$

Furthermore, we assume the function $g:\Omega \times \mathbb{R}\to \mathbb{R}$ is continuous and verifies the following two conditions.

$\left(G_1\right)$There exist $c_1>0$ and $q\in C\left(\overline{\Omega }\right)$ satisfying

$$\left|g(x,t)\right|\le c_1{\left|t\right|}^{q\left(x\right)-1},\mathit{for}\mathit{all}(x,t)\in \Omega \times \mathbb{R}$$

and

$$\beta p^{+}<q^{-}=\underset{x\in \overline{\Omega }}{inf}q\left(x\right)<q\left(x\right)<p_s^{*}\left(x\right),\mathit{for}\mathit{all}x\in \Omega ,$$

where β is given in $\left(M_1\right)$ above.

$\left(G_2\right)$For $h_1$, $h_2$, and β given in $\left(M_1\right)$, there exist $t_0$ and $\lambda \in \left({\displaystyle \frac{h_2\beta {\left(p^{+}\right)}^{\beta }}{h_1{\left(p^{-}\right)}^{\beta -1}},+\infty }\right)$, such that

$$0<\lambda G(x,t)\le tg(x,t),\mathrm{for}\mathrm{all}t\in \mathbb{R}\mathrm{with}\left|t\right|\ge t_0,\mathit{and}x\in \Omega ,$$

where $G(x,t)={\int }_0^{t}g(x,s)ds.$

Furthermore, we propose the following condition on the function g.

$\left(G_3\right):$$g(x,-t)=-g(x,t)$ for all $(x,t)\in \Omega \times \mathbb{R}$.

In the operator ${(-\Delta )}_{p(·)}^{s(·)}$, we suppose that continuous functions $s(·):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (0,1)$ and $p(·):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (1,\infty )$ fulfill

(H₁):

$0<s^{-}\le s^{+}<1<p^{-}\le p^{+}$;

(H₂):

$s(·)$ and $p(·)$ are symmetric, i.e., $s(x,y)=s(y,x)$ and $p(x,y)=p(y,x)$ for any $(x,y)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$.

Clearly, the operator in (2) reduces to the fractional p-Laplacian ${(-\Delta )}_p^s$ as $p(x,y)\equiv p$ and $s(x,y)\equiv s$; see [6,7,8], and the references therein. In particular, we point out that An et al. in [9] studied the existence of infinitely many solutions for a class of fractional p-Laplacian equations by using the fountain theorem. They also investigated a fractional p-Laplacian system with the help of the Nehari manifold method in [10]. This type of operator has a widespread application in various sciences, such as mechanics [11], finance [12], and so on. For a Kirchhoff situation, we recall [13] where the authors construct a stationary fractional Kirchhoff problem, which is excellent pioneering work. It is worth noting that a typical non-local equation that has attracted attention is the Kirchhoff type equation, which is a physical model given by Kirchhoff [14] in 1883, i.e.,

$$\rho \frac{{\partial }^{2}u}{\partial t^2}-\left(\frac{P_0}{h}+\frac{E}{2L}{\int }_0^L{\left|\frac{\partial u}{\partial x}\right|}^{2}dx\right)\frac{{\partial }^{2}u}{\partial x^2}=0,$$

(3)

where u denotes the displacement of the string, $\rho$ denotes the mass density, $P_0$ denotes the initial tension, h denotes the area of the cross section, E denotes the Young’s modulus of the material, and L denotes the length of the string. For more physical phenomena described by classical Kirchhoff theory, see [15].

In addition, in the scope of ordinary differential equation research, non-local problems have also received extensive attention, and we specifically point out two excellent studies [16,17].

Very recently, great interest has been devoted to the investigation of fractional problems involving possibly variable order and variable exponent. The classic example is from Chen, Levine, and Rao [18], and it concerns applications to image restoration. We also refer to [19,20] for a multiplicity result for a problem driven by ${(-\Delta )}^{s(·)}$, that is, operator (2) with $p(x,y)\equiv 2$. In [21,22,23,24], different approaches are described to handle a fractional operator ${(-\Delta )}_{p(·)}^s$, with $s(x,y)\equiv s$. Papers [25,26,27,28] introduce variational techniques and properties for the local version of operator ${(-\Delta )}_{p(·)}^s$, that is, with the integral in (2) set on $\Omega$ instead of ${\mathbb{R}}^N$. Finally, the authors in [29,30] try to consider problems involving a variable-order fractional operator with variable exponent $p(·)$.

Motivated by the above papers, we study a new double variable order fractional Kirchhoff type problem (1). As far as we know, very few papers have studied the infinite number of solutions to such a bi-non-local equation. Indeed, in [22], the authors considered a class of fractional $p(·)$-Kirchhoff type problems, such as (1) but with $s(x,y)\equiv s$, $\mu =0$ and g of a model form. Thus, our main results stated below generalize ([22], Theorems 3.1 and 3.3) in several directions, and somehow the existence results in [21,24].

Theorem 1.

If the conditions $\left(H_1\right)-\left(H_2\right)$, $\left(M_1\right)$, and $\left(G_1\right)-\left(G_2\right)$ hold, then, there exists ${\mu }^{*}>0$ such that for any $\mu \in (-\infty ,{\mu }^{*}]$ problem (1) admits a non-trivial weak solution.

By further assuming the symmetric condition $\left(G_3\right)$, we obtain the following multiplicity result.

Theorem 2.

If the conditions $\left(H_1\right)-\left(H_2\right)$, $\left(M_1\right)$, and $\left(G_1\right)-\left(G_3\right)$ hold, then, for any $\mu \in \mathbb{R}$ problem (1) has infinitely many weak solutions with unbounded positive energy.

The remaining sections are organized as follows. Section 2 introduces some lemmas and knowledge of space theory. Section 3 verifies the Palais–Smale condition. Section 4 gives the proof of Theorem 1. Section 5 proves Theorem 2.

2. Functional Analytic Setup and Preliminaries

Let

$$C_{+}\left(\overline{\Omega }\right)=\left\{r\in C\left(\overline{\Omega }\right):1<r\left(x\right)\mathrm{for}\mathrm{all}x\in \overline{\Omega }\right\}.$$

For any $r\in C_{+}\left(\overline{\Omega }\right)$ we recall the variable exponent Lebesgue space

$$L^{r(·)}(\Omega )=\left\{u:\mathrm{the}\mathrm{function}u:\Omega \to \mathbb{R}\mathrm{is}\mathrm{measurable},{\int }_{\Omega }{\left|u\left(x\right)\right|}^{r\left(x\right)}dx<\infty \right\},$$

with the norm

$${\parallel u\parallel }_{r(·)}=inf\left\{\gamma >0:{\int }_{\Omega }|\frac{u\left(x\right)}{\gamma }{|}^{r\left(x\right)}dx\le 1\right\}.$$

Then $(L^{r(·)}(\Omega ),\parallel ·{\parallel }_{r(·)})$ is a separable reflexive Banach space (see [31], Theorem 2.5 and Corollaries 2.7 and 2.12).

Let $\tilde{r}\in C_{+}\left(\overline{\Omega }\right)$ be the conjugate exponent of r, that is

$$\frac{1}{r\left(x\right)}+\frac{1}{\tilde{r}\left(x\right)}=1,\mathrm{for}\mathrm{all} x\in \overline{\Omega }.$$

Then we have the following Hölder inequality (see [31], Theorem 2.1).

Lemma 1.

Suppose that $u\in L^{r(·)}(\Omega )$ and $v\in L^{\tilde{r}(·)}(\Omega )$. Then

$$\left|{\int }_{\Omega }uvdx\right|\le \left(\frac{1}{r^{-}}+\frac{1}{{\tilde{r}}^{-}}\right){\parallel u\parallel }_{r(·)}{\parallel v\parallel }_{\tilde{r}(·)}\le {2\parallel u\parallel }_{r(·)}{\parallel v\parallel }_{\tilde{r}(·)}.$$

Defining the modular functional ${\rho }_{r(·)}:L^{r(·)}(\Omega )\to \mathbb{R}$, by

$${\rho }_{r(·)}\left(u\right)={\int }_{\Omega }{\left|u\left(x\right)\right|}^{r\left(x\right)}dx,$$

we have the next crucial result given in [32].

Proposition 1.

Suppose that $u\in L^{r(·)}(\Omega )$ and $\left\{u_j\right\}\subset L^{r(·)}(\Omega )$. Then

(1) 

${\parallel u\parallel }_{r(·)}<1(resp.=1,>1)\iff {\rho }_{r(·)}\left(u\right)<1(resp.=1,>1),$

(2) 

${\parallel u\parallel }_{r(·)}<1\Rightarrow {\parallel u\parallel }_{r(·)}^{r^{+}}\le {\rho }_{r(·)}\left(u\right)\le {\parallel u\parallel }_{r(·)}^{r^{-}},$

(3) 

${\parallel u\parallel }_{r(·)}>1\Rightarrow {\parallel u\parallel }_{r(·)}^{r^{-}}\le {\rho }_{r(·)}\left(u\right)\le {\parallel u\parallel }_{r(·)}^{r^{+}},$

(4) 

$\underset{j\to \infty }{lim}{\parallel u_j\parallel }_{r(·)}=0(\infty )\iff \underset{j\to \infty }{lim}{\rho }_{r(·)}\left(u_j\right)=0(\infty ),$

(5) 

$\underset{j\to \infty }{lim}{\parallel u_j-u\parallel }_{r(·)}=0\iff \underset{j\to \infty }{lim}{\rho }_{r(·)}(u_j-u)=0.$

The variable order fractional Sobolev spaces with variable exponent is defined by

$$W^{s(·),p(·)}(\Omega )=\left\{u\in L^{\overline{p}(·)}(\Omega ):{\int \int }_{\Omega \times \Omega }\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{\gamma }^{p(x,y)}{|x-y|}^{N+p(x,y)s(x,y)}}dxdy<\infty ,\mathrm{for}\mathrm{some}\gamma >0\right\}$$

with the norm ${\parallel u\parallel }_{s(·),p(·)}={\parallel u\parallel }_{\overline{p}(·)}+{\left[u\right]}_{s(·),p(·)}$, where

$${\left[u\right]}_{s(·),p(·)}=inf\left\{\gamma >0:{\int \int }_{\Omega \times \Omega }\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{\gamma }^{p(x,y)}{|x-y|}^{N+p(x,y)s(x,y)}}dxdy<1\right\}.$$

We define the new variable order fractional Sobolev spaces with variable exponent (see more details in reference [29]):

$$X=\left\{u:{\mathbb{R}}^N{\to \mathbb{R}:u|}_{\Omega }\in L^{\overline{p}(·)}(\Omega ),{\int \int }_Q\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{\gamma }^{p(x,y)}{|x-y|}^{N+p(x,y)s(x,y)}}dxdy<\infty ,\mathrm{for}\mathrm{some}\gamma >0\right\},$$

where $Q:={\mathbb{R}}^{2N}\setminus ({\Omega }^c\times {\Omega }^c)$. The space X is endowed with the norm

$${\parallel u\parallel }_X={\parallel u\parallel }_{\overline{p}(·)}+{\left[u\right]}_X,$$

where

$${\left[u\right]}_X=inf\left\{\gamma >0:{\int \int }_Q\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{\gamma }^{p(x,y)}{|x-y|}^{N+p(x,y)s(x,y)}}dxdy<1\right\}.$$

We notice that the norms ${\parallel ·\parallel }_{s(·),p(·)}$ and ${\parallel ·\parallel }_X$ are not the same because $\Omega \times \Omega \subset Q$ and $\Omega \times \Omega \ne Q$. This makes $W^{s(·),p(·)}(\Omega )$ not sufficient for studying the kind of problem like (1).

For this, we set our Banach space workspace as

$$X_0=\left\{u\in X:u=0a.e.\mathrm{in}{\mathbb{R}}^N\setminus \Omega \right\},$$

which is separable and reflexive (see [30], Proposition 3.7), with respect to the norm

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{X_0}& =inf\left\{\gamma >0:{\int \int }_Q\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{\gamma }^{p(x,y)}{|x-y|}^{N+p(x,y)s(x,y)}}dxdy<1\right\}\hfill \\ \hfill & =inf\left\{\gamma >0:{\int \int }_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{\gamma }^{p(x,y)}{|x-y|}^{N+p(x,y)s(x,y)}}dxdy<1\right\},\hfill \end{array}$$

where the last equality is a consequence of the fact that $u=0$ a.e. in ${\mathbb{R}}^N\setminus \Omega$.

We are ready to introduce an embedding theorem for $X_0$, given in ([29], Theorem 2.5).

Lemma 2.

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded smooth domain. Let $p(·)$ and $s(·)$ satisfy $\left(H_1\right)-\left(H_2\right)$, such that $N>p(x,y)s(x,y)$ for all $(x,y)\in \overline{\Omega }\times \overline{\Omega }$. Then for any $r\in C_{+}\left(\overline{\Omega }\right)$ with $1<r\left(x\right)<p_s^{*}\left(x\right)$ for $x\in \overline{\Omega }$, there exists a positive constant $C_r=C_r(N,s,p,r,\Omega )$ such that

$${\parallel u\parallel }_{r\left(x\right)}\le C_r{\parallel u\parallel }_{X_0}$$

(4)

for any $v\in X_0$. Furthermore, the embedding $X_0\hookrightarrow L^{r(·)}(\Omega )$ is compact.

We note that ${\parallel ·\parallel }_{X_0}$ and ${\parallel ·\parallel }_X$ are equivalent norms on $X_0$. We define the fractional modular functional ${\varrho }_{p(·)}^{s(·)}:X_0\to \mathbb{R}$, by

$${\varrho }_{p(·)}^{s(·)}\left(u\right)={\int \int }_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy.$$

Then, similar to Proposition 1, we get

Proposition 2.

([30], Lemmas 3.4 and 3.5). Suppose that $u\in X_0$ and $\left\{u_j\right\}\subset X_0$. Then

(1) 

${\parallel u\parallel }_{X_0}<1(resp.=1,>1)\iff {\varrho }_{p(·)}^{s(·)}\left(u\right)<1(resp.=1,>1),$

(2) 

${\parallel u\parallel }_{X_0}<1\Rightarrow {\parallel u\parallel }_{X_0}^{p^{+}}\le {\varrho }_{p(·)}^{s(·)}\left(u\right)\le {\parallel u\parallel }_{X_0}^{p^{-}},$

(3) 

${\parallel u\parallel }_{X_0}>1\Rightarrow {\parallel u\parallel }_{X_0}^{p^{-}}\le {\varrho }_{p(·)}^{s(·)}\left(u\right)\le {\parallel u\parallel }_{X_0}^{p^{+}},$

(4) 

$\underset{j\to \infty }{lim}{\parallel u_j\parallel }_{X_0}=0(\infty )\iff \underset{j\to \infty }{lim}{\varrho }_{p(·)}^{s(·)}\left(u_j\right)=0(\infty ),$

(5) 

$\underset{j\to \infty }{lim}{\parallel u_j-u\parallel }_{X_0}=0\iff \underset{j\to \infty }{lim}{\varrho }_{p(·)}^{s(·)}(u_j-u)=0.$

A function $u\in X_0$ is a weak solution of problem (1), if

$$\begin{array}{c}\hfill M\left({\delta }_{p(·)}\left(u\right)\right){\int \int }_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)-2}(u\left(x\right)-u\left(y\right))(\varphi \left(x\right)-\varphi \left(y\right))}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy\\ \hfill =\mu {\int }_{\Omega }{\left|u\left(x\right)\right|}^{\overline{p}\left(x\right)-2}u\left(x\right)\varphi \left(x\right)dx+{\int }_{\Omega }g(x,u)\varphi dx,\end{array}$$

(5)

for all $\varphi \in X_0$, where

$${\delta }_{p(·)}\left(u\right)={\int \int }_{{\mathbb{R}}^{2N}}\frac{1}{p(x,y)}\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)}}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy.$$

Considering the following functional associated with problem (1), defined by ${\mathcal{I}}_{\mu }:X_0\to \mathbb{R}$

$${\mathcal{I}}_{\mu }\left(u\right)=\tilde{M}\left({\delta }_{p(·)}\left(u\right)\right)-\mu {\int }_{\Omega }\frac{1}{\overline{p}\left(x\right)}{\left|u\left(x\right)\right|}^{\overline{p}\left(x\right)}dx-{\int }_{\Omega }G(x,u)dx,$$

where $\tilde{M}\left(t\right)={\int }_0^{t}M\left(\tau \right)d\tau .$ Clearly, it follows from the continuity of M that ${\mathcal{I}}_{\mu }$ is well defined and of class $C^1$ on $X_0$. Furthermore, we have

$$\begin{array}{cc}\hfill \langle {\mathcal{I}}_{\mu }^{{}^{\prime }}\left(u\right),\varphi \rangle =& M\left({\delta }_{p(·)}\left(u\right)\right){\int \int }_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)-2}(u\left(x\right)-u\left(y\right))(\varphi \left(x\right)-\varphi \left(y\right))}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy\hfill \\ \hfill & -\mu {\int }_{\Omega }{\left|u\left(x\right)\right|}^{\overline{p}\left(x\right)-2}u\left(x\right)\varphi \left(x\right)dx-{\int }_{\Omega }g(x,u)\varphi \left(x\right)dx,\hfill \end{array}$$

for all u, $\varphi \in X_0$. Hence, the weak solutions of problem (1) are the critical points of ${\mathcal{I}}_{\mu }$. If such a weak solution exists and is non-trivial, then $\mu$ is an eigenvalue of problem (1).

We conclude this section by presenting a technical result that is useful in studying the compactness of ${\mathcal{I}}_{\mu }$. The proof of this proposition is similar to ([26], Lemma 4.2) and working on $X_0$.

Proposition 3.

We consider the following functional $\mathcal{A}:X_0\to X_0^{*}$, with $X_0^{*}$ the dual space of $X_0$, such that

$$\langle \mathcal{A}\left(u\right),\varphi \rangle ={\int \int }_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)-2}(u\left(x\right)-u\left(y\right))(\varphi \left(x\right)-\varphi \left(y\right))}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy,$$

for any u, $\varphi \in X_0$. Then:

(i) 

The operator $\mathcal{A}$ is bounded and strictly monotone;

(ii) 

$\mathcal{A}$ is a mapping of type $\left(S_{+}\right)$, that is, if $u_j\rightharpoonup u\in X_0$ and $\underset{j\to \infty }{lim\; sup}\mathcal{A}\left(u_j\right)(u_j-u)\le 0,$ then $u_j\to u\in X_0$;

(iii) 

$\mathcal{A}:X_0\to X_0^{*}$ is a homeomorphism.

Throughout this paper, for simplicity, we use $\{c_i,i\in \mathbb{N}\}$ to denote different non-negative or positive constants. In addition, we denote with $c^{+}$ and $c^{-}$, respectively, the positive part and negative part of a number $c\in \mathbb{R}$.

3. Palais–Smale Condition

We now recall the definition of the Palais–Smale condition. We say that ${\mathcal{I}}_{\mu }$ fulfills the Palais–Smale condition at the level $c\in \mathbb{R}$ if any sequence $u_j\subset X_0$ fulfilling

$${\mathcal{I}}_{\mu }\left(u_j\right)\to c\mathrm{and}{\mathcal{I}}_{\mu }^{{}^{\prime }}\left(u_j\right)\to 0\mathrm{in}{X}_0^{*}\mathrm{as}j\to \infty ,$$

(6)

possesses a convergent subsequence in $X_0$.

Lemma 3.

Suppose that $\left(M_1\right)$, $\left(G_1\right)-\left(G_2\right)$, and $\left(H_1\right)-\left(H_2\right)$ hold. Then for any $\mu \in \mathbb{R}$ the functional ${\mathcal{I}}_{\mu }$ fulfills the Palais–Smale condition for any $c\in \mathbb{R}$.

Proof. 

Let $\mu \in \mathbb{R}$. Suppose a sequence $\left\{u_j\right\}\subset X_0$ verifying (6). We argue in two steps.

Step 1. We first show that the sequence $\left\{u_j\right\}$⊂$X_0$ is bounded. For this end, by $\left(M_1\right)$, $\left(G_2\right)$, Propositions 1 and 2, and Lemma 2, we get

$$\begin{array}{cc}\hfill \lambda {\mathcal{I}}_{\mu }\left(u_j\right)-\langle {\mathcal{I}}_{\mu }^{{}^{\prime }}\left(u_j\right),u_j\rangle =& \lambda \tilde{M}\left({\delta }_{p(·)}\left(u_j\right)\right)-M\left({\delta }_{p(·)}\left(u_j\right)\right){\int \int }_{{\mathbb{R}}^{2N}}\frac{|u_j\left(x\right)-u_j{\left(y\right)|}^{p(x,y)}}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy\hfill \\ \hfill & -\mu {\int }_{\Omega }\left(\frac{\lambda }{\overline{p}\left(x\right)}-1\right){\left|u_j\right|}^{\overline{p}\left(x\right)}dx-{\int }_{\Omega }\left(\lambda G(x,u_j)-g(x,u_j)u_j\right)dx\hfill \\ \hfill & \ge \frac{\lambda h_1}{\beta }{\left({\delta }_{p(·)}\left(u_j\right)\right)}^{\beta }-h_2{\left({\delta }_{p(·)}\left(u_j\right)\right)}^{\beta -1}\left({\varrho }_{p(·)}^{s(·)}\left(u_j\right)\right)-{\mu }^{+}{\int }_{\Omega }\left(\frac{\lambda }{\overline{p}\left(x\right)}-1\right){\left|u_j\right|}^{\overline{p}\left(x\right)}dx\hfill \\ \hfill & \ge \frac{\lambda h_1}{\beta {\left(p^{+}\right)}^{\beta }}{\left({\varrho }_{p(·)}^{s(·)}\left(u_j\right)\right)}^{\beta }-\frac{h_2}{{\left(p^{-}\right)}^{\beta -1}}{\left({\varrho }_{p(·)}^{s(·)}\left(u_j\right)\right)}^{\beta }-{\mu }^{+}\left(\frac{\lambda }{p^{-}}-1\right){\varrho }_{\overline{p}(·)}\left(u_j\right)\hfill \end{array}$$

$$\ge \left(\frac{\lambda h_1}{\beta {\left(p^{+}\right)}^{\beta }}-\frac{h_2}{{\left(p^{-}\right)}^{\beta -1}}\right)min\left\{\parallel u_j{\parallel }_{X_0}^{\beta p^{-}},{\parallel u_j\parallel }_{X_0}^{\beta p^{+}}\right\}-{\mu }^{+}\left(\frac{\lambda }{p^{-}}-1\right)max\left\{(C_{\overline{p}}\parallel u_j{\parallel }_{X_0}{)}^{{\overline{p}}^{-}},(C_{\overline{p}}\parallel u_j{{\parallel }_{X_0})}^{{\overline{p}}^{+}}\right\},$$

and recall that $\lambda >p^{+}\ge \overline{p}\left(x\right)\ge p^{-}$ for $x\in \overline{\Omega }$, by $\left(G_2\right)$. Thus from (6), there exists ${\sigma }_{\mu }>0$ such that as $j\to \infty$, there holds

$$\lambda c+{\sigma }_{\mu }{\parallel u_j\parallel }_{X_0}+o\left(1\right)\ge \left(\frac{\lambda h_1}{\beta {\left(p^{+}\right)}^{\beta }}-\frac{h_2}{{\left(p^{-}\right)}^{\beta -1}}\right)min\left\{\parallel u_j{\parallel }_{X_0}^{\beta p^{-}},{\parallel u_j\parallel }_{X_0}^{\beta p^{+}}\right\}-{\mu }^{+}\left(\frac{\lambda }{p^{-}}-1\right)max\left\{(C_{\overline{p}}\parallel u_j{\parallel }_{X_0}{)}^{{\overline{p}}^{-}},(C_{\overline{p}}\parallel u_j{{\parallel }_{X_0})}^{{\overline{p}}^{+}}\right\},$$

which implies that $\left\{u_j\right\}$ is bounded in $X_0$, as $1<p^{-}\le {\overline{p}}^{-}\le {\overline{p}}^{+}\le p^{+}<\beta p^{-}\le \beta p^{+}$ by $\left(G_1\right)$.

Step 2. We will show that $\left\{u_j\right\}$ converges strongly in $X_0$. In view of Lemma 2 and the reflexivity of $X_0$, that there exists a subsequence, still denoted by $\left\{u_j\right\}$, and $u\in X_0$ such that

$$u_j\rightharpoonup u\mathrm{in}{X}_0,u_j\to u\mathrm{in}{L}^{r(·)}(\Omega ),u_j\left(x\right)\to u\left(x\right)\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,$$

(7)

for any $r\in C_{+}\left(\overline{\Omega }\right)$, with $1<r\left(x\right)<p_s^{*}\left(x\right)$ for $x\in \overline{\Omega }$. Using the Hölder inequality (Lemma 1) and (7) with $r\equiv \overline{p}$, from $p^{+}<\beta p^{+}<p_s^{*}\left(x\right)$ for $x\in \overline{\Omega }$, by $\left(G_1\right)$, we get

$$\left|{\int }_{\Omega }|u_j{|}^{\overline{p}\left(x\right)-2}{u}_j(u_j-u)dx\right|\le {\int }_{\Omega }|u_j{|}^{\overline{p}\left(x\right)-1}|u_j-u|dx\le 2\parallel |u_j{|}^{\overline{p}\left(x\right)-1}{\parallel }_{\frac{\overline{p}\left(x\right)}{\overline{p}\left(x\right)-1}}{\parallel u_j-u\parallel }_{\overline{p}\left(x\right)}\to 0\mathrm{as}j\to \infty .$$

Therefore,

$$\underset{j\to \infty }{lim}{\int }_{\Omega }{\left|u_j\right|}^{\overline{p}\left(x\right)-2}{u}_j(u_j-u)dx=0.$$

(8)

According to $\left(G_1\right)$, (7) with $r\equiv q$ and the Hölder inequality (Lemma 1), we have

$$\left|{\int }_{\Omega }g(x,u_j)(u_j-u)dx\right|\le c_1{\int }_{\Omega }|u_j{|}^{q\left(x\right)-1}|u_j-u|dx\le 2c_1\parallel |u_j{|}^{q\left(x\right)-1}{\parallel }_{\frac{q\left(x\right)}{q\left(x\right)-1}}{\parallel u_j-u\parallel }_{q\left(x\right)}\to 0\mathrm{as}j\to \infty ,$$

which implies that

$$\underset{j\to \infty }{lim}{\int }_{\Omega }g(x,u_j)(u_j-u)dx=0.$$

(9)

By virtue of (6), we get

$$\langle {\mathcal{I}}_{\mu }^{{}^{\prime }}\left(u_j\right),u_j-u\rangle \to 0.$$

(10)

As $\left\{u_j\right\}$ is bounded in $X_0$, and in view of Proposition 2, passing to subsequence, if necessary, we suppose that

$${\delta }_{p(·)}\left(u_j\right)\to \kappa \ge 0,\mathrm{as}j\to \infty .$$

If $\kappa =0$, then $\left\{u_j\right\}$ strongly converges to $u=0$ in $X_0$ and the proof is complete.

If $\kappa >0$, in view of the function M is continuous, we know

$$M\left({\delta }_{p(·)}\left(u_j\right)\right)\to M\left(\kappa \right)>0\mathrm{as}j\to \infty .$$

Thus, it follows from $\left(M_1\right)$ that

$$0<c_2<M\left({\delta }_{p(·)}\left(u_j\right)\right)<c_3\mathrm{as}j\to \infty .$$

(11)

From (8)–(11), we obtain

$$\underset{j\to \infty }{lim}{\int \int }_{{\mathbb{R}}^{2N}}\frac{|u_j\left(x\right)-u_j{\left(y\right)|}^{p(x,y)-2}\left(u_j\left(x\right)-u_j\left(y\right)\right)\left((u_j\left(x\right)-u_j\left(y\right))-(u\left(x\right)-u\left(y\right))\right)}{{|x-y|}^{N+p(x,y)s(x,y)}}dxdy=0.$$

Now together with (7), we have

$$u_j\rightharpoonup u\in X_0,\underset{j\to \infty }{lim\; sup}\mathcal{A}\left(u_j\right)(u_j-u)\le 0.$$

Therefore, $\mathcal{A}\mathrm{is}\mathrm{a}\mathrm{mapping}\mathrm{of}\mathrm{type}\left(S_{+}\right)$, which implies that $u_j\to u$ in $X_0$ from Proposition 3. This concludes the proof of the Palais–Smale compactness condition.

□

4. Proof of Theorem 1

The next two lemmas verify the mountain pass geometry of ${\mathcal{I}}_{\mu }$.

Lemma 4.

Suppose that $\left(M_1\right)$, $\left(G_1\right)$, and $\left(H_1\right)-\left(H_2\right)$ hold. Then, there exist numbers $\rho >0$, ${\mu }^{*}={\mu }^{*}\left(\rho \right)>0$ and $\alpha =\alpha \left(\rho \right)>0$ such that ${\mathcal{I}}_{\mu }\left(u\right)\ge \alpha >0$ for any $u\in X_0$ with ${\parallel u\parallel }_{X_0}=\rho$, and for any $\mu \in (-\infty ,{\mu }^{*}]$.

Proof. 

Let $u\in X_0$ be such that ${\parallel u\parallel }_{X_0}=\rho \in \left(0,min\left\{1,1/C_{\overline{p}},1/C_q\right\}\right)$, with $C_{\overline{p}}$ and $C_q$ given in Lemma 2. In view of $\left(G_1\right)$, Propositions 1 and 2, and Lemma 2, we have that

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mu }\left(u\right)& \ge \tilde{M}\left({\delta }_{p(·)}\left(u\right)\right)-{\mu }^{+}{\int }_{\Omega }\frac{1}{\overline{p}\left(x\right)}{\left|u\left(x\right)\right|}^{\overline{p}\left(x\right)}dx-c_1{\int }_{\Omega }\frac{1}{q\left(x\right)}{\left|u\left(x\right)\right|}^{q\left(x\right)}dx\hfill \\ \hfill & \ge \frac{h_1}{\beta }{\left({\delta }_{p(·)}\left(u\right)\right)}^{\beta }-\frac{{\mu }^{+}}{p^{-}}{\rho }_{\overline{p}(·)}\left(u\right)-\frac{c_1}{q^{-}}{\rho }_{q(·)}\left(u\right)\hfill \\ \hfill & \ge \frac{h_1}{\beta {\left(p^{+}\right)}^{\beta }}{\parallel u\parallel }_{X_0}^{\beta p^{+}}-\frac{{\mu }^{+}}{p^{-}}{C}_{\overline{p}}^{{\overline{p}}^{-}}{\parallel u\parallel }_{X_0}^{p^{-}}-\frac{c_1}{q^{-}}{C}_q^{q^{-}}{\parallel u\parallel }_{X_0}^{q^{-}}\hfill \\ \hfill & ={\rho }^{\beta p^{+}}\left(\frac{h_1}{\beta {\left(p^{+}\right)}^{\beta }}-\frac{c_1{C}_q^{q^{-}}}{q^{-}}{\rho }^{q^{-}-\beta p^{+}}\right)-\frac{{\mu }^{+}{C}_{\overline{p}}^{{\overline{p}}^{-}}}{p^{-}}{\rho }^{p^{-}}.\hfill \end{array}$$

Let us consider

$$\tilde{\rho }={\left(\frac{h_1}{2\beta {\left(p^{+}\right)}^{\beta }}·\frac{q^{-}}{c_1{C}_q^{q^{-}}}\right)}^{\frac{1}{q^{-}-\beta p^{+}}}\mathrm{and}{\mu }^{*}=\frac{p^{-}}{C_{\overline{p}}^{{\overline{p}}^{-}}}·\frac{h_1}{4\beta {\left(p^{+}\right)}^{\beta }}{\rho }^{\beta p^{+}-p^{-}}.$$

Then, for any $u\in X_0$ with ${\parallel u\parallel }_{X_0}=\rho \in \left(0,min\left\{1,1/C_{\overline{p}},1/C_q,\tilde{\rho }\right\}\right)$ and all $\mu \in (-\infty ,{\mu }^{*}]$, since $\beta p^{+}<q^{-}$ by $\left(G_1\right)$ we have

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mu }\left(u\right)& \ge {\rho }^{\beta p^{+}}\left(\frac{h_1}{\beta {\left(p^{+}\right)}^{\beta }}-\frac{c_1{C}_q^{q^{-}}}{q^{-}}{\tilde{\rho }}^{q^{-}-\beta p^{+}}\right)-\frac{2{\mu }^{+}{C}_{\overline{p}}^{{\overline{p}}^{-}}}{p^{-}}{\rho }^{p^{-}}\hfill \\ \hfill & =\frac{h_1}{2\beta {\left(p^{+}\right)}^{\beta }}{\rho }^{\beta p^{+}}-\frac{{\mu }^{+}{C}_{\overline{p}}^{{\overline{p}}^{-}}}{p^{-}}{\rho }^{p^{-}}\ge \frac{h_1}{4\beta {\left(p^{+}\right)}^{\beta }}{\rho }^{\beta p^{+}}=\alpha >0,\hfill \end{array}$$

concluding the proof. □

Lemma 5.

Suppose that $\left(M_1\right)$, $\left(G_2\right)$, and $\left(H_1\right)-\left(H_2\right)$ hold. Then, for any $\mu \in \mathbb{R}$ there exists $u\in X_0$ with ${\parallel u\parallel }_{X_0}>\rho$, where $\rho >0$ is given in Lemma 4, such that ${\mathcal{I}}_{\mu }\left(u\right)<0$.

Proof. 

Let $\mu \in \mathbb{R}$. By $\left(G_2\right)$, we have that for all $D>0$, there exists $C_D>0$ such that

$$G(x,t)\ge {D\left|t\right|}^{\lambda }-C_D,\mathrm{for}\mathrm{all}(x,t)\in \Omega \times \mathbb{R}.$$

(12)

Take $\phi \in C_0^{\infty }\left({\mathbb{R}}^N\right)$, with $\phi >0$. Let $t>1$. From (12) and $\left(M_1\right)$ we get

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mu }\left(t\phi \right)& =\tilde{M}\left({\delta }_{p(·)}\left(t\phi \right)\right)-\mu {\int }_{\Omega }\frac{1}{\overline{p}\left(x\right)}{\left|t\phi \right|}^{\overline{p}\left(x\right)}dx-{\int }_{\Omega }G(x,t\phi )dx\hfill \\ \hfill & \le \frac{h_2}{\beta {\left(p^{-}\right)}^{\beta }}{t}^{\beta p^{+}}{\left({\varrho }_{p(·)}^{s(·)}\left(\phi \right)\right)}^{\beta }-\frac{{\mu }^{-}}{p^{+}}{t}^{p^{-}}{\int }_{\Omega }{\left|\phi \right|}^{\overline{p}\left(x\right)}dx-D{t}^{\lambda }{\int }_{\Omega }{\left|\phi \right|}^{\lambda }dx+C_D|\Omega |.\hfill \end{array}$$

Since $h_2\ge h_1$ and $p^{+}\ge p^{-}$, we get $\lambda >\beta p^{+}\ge \beta p^{-}>p^{-}$, we deduce that ${\mathcal{I}}_{\mu }\left(t\phi \right)\to -\infty$ as $t\to \infty$. Then for $t>1$ sufficiently large, we can let $u=t\phi$ such that ${\parallel u\parallel }_{X_0}>\rho$ and ${\mathcal{I}}_{\mu }\left(u\right)<0$. □

Proof of Theorem 1.

According to Lemmas 3–5, considering also that ${\mathcal{I}}_{\mu }\left(0\right)=0$, our functional ${\mathcal{I}}_{\mu }$ fulfills all conditions of the mountain pass theorem. Thus, problem (1) has a non-trivial weak solution. □

5. Proof of Theorem 2

The proof of Theorem 2 is based on the application of the fountain theorem, which can be found in [33]. For this, as the real Banach space $X_0$ is reflexive and separable, there exist $\left\{w_i\right\}\subset X_0$ and $\left\{w_i^{*}\right\}\subset X_0^{*}$ such that

$$X_0=\overline{\mathrm{span}\{w_i:i\in {\mathbb{N}}^{+}\}},X_0^{*}=\overline{\mathrm{span}\{w_i^{*}:i\in {\mathbb{N}}^{+}\}}$$

and

$$\langle w_i^{*},w_j\rangle =\left\{\begin{array}{c}1,i=j,\hfill \\ 0,i\ne j.\hfill \end{array}\right.$$

$$X_0^i=\mathrm{span}\left\{w_i\right\},Y_j=\underset{i=1}{\overset{j}{⨁}}{X}_0^i,Z_j=\overline{\underset{i=j}{\overset{\infty }{⨁}}{X}_0^i},j=1,2,\dots$$

Now we are ready to introduce the fountain theorem.

Theorem 3.

([33]) Consider an even functional $I\in C^1(X_0,\mathbb{R})$. Assume that for every $j\in \mathbb{N}$, there exist ${\rho }_j>{\gamma }_j>0$ such that

(I₁)

$a_j:=\underset{u\in Y_j,{\parallel u\parallel }_{X_0}={\rho }_j}{max}I\left(u\right)\le 0,$

(I₂)

$b_j:=\underset{u\in Z_j,{\parallel u\parallel }_{X_0}={\gamma }_j}{inf}I\left(u\right)\to +\infty ,j\to \infty ,$

(I₃)

I fulfills the Palais–Smale condition for every $c>0$.

Then I has an unbounded sequence of critical values.

Lemma 6.

Suppose that $\left(H_1\right)$–$\left(H_2\right)$ hold. Let $r\in C_{+}\left(\overline{\Omega }\right)$, with $1<r\left(x\right)<p_s^{*}\left(x\right)$ for any $x\in \overline{\Omega }$, and denote

$${\xi }_j:=sup\left\{{\parallel u\parallel }_{r(·)}:u\in Z_j,{\parallel u\parallel }_{X_0}=1\right\}.$$

(13)

Then, ${\xi }_j\to 0$ as $j\to \infty$

Proof. 

By definition of $Z_j$ we have $Z_{j+1}\subset Z_j$ and so $0<{\xi }_{j+1}\le {\xi }_j$ for any $j\in \mathbb{N}$. Thus ${\xi }_j\to \xi \ge 0$ as $j\to \infty$. Moreover, by (13) there exists $v_j\in Z_j$ such that

$$\parallel u_j{\parallel }_{X_0}=1,0\le {\xi }_j-{\parallel u_j\parallel }_{r(·)}<\frac{1}{j}.$$

(14)

As $\left\{u_j\right\}$ is bounded in $X_0$, there exists a subsequence of $\left\{u_j\right\}$, still denoted by $u_j$, such that $u_j\rightharpoonup u$ in $X_0$ and $\langle w_i^{*},u\rangle =\underset{j\to \infty }{lim}\langle w_i^{*},u_j\rangle =0$ for $i\in {\mathbb{N}}^{+}.$ Hence we have $u=0$. Furthermore, by Lemma 2 we obtain that $u_j\to 0$ in $L^{r(·)}(\Omega )$. Therefore, by (14) we have ${\xi }_j\to 0$ as $j\to \infty$. □

Proof of Theorem 2.

We know that the functional ${\mathcal{I}}_{\mu }$ fulfills the Palais–Smale condition by Lemma 3. In what follows, we show that ${\mathcal{I}}_{\mu }$ satisfies all conditions of Theorem 3, step by step.

In view of $\left(G_1\right)$ and $\left(G_2\right)$, there exist two positive numbers $c_4$ and $c_5$ such that

$$\left|G(x,t)\right|\ge c_4{\left|t\right|}^{\lambda }-c_5\left|t\right|,\mathrm{for}\mathrm{all}(x,t)\in \Omega \times \mathbb{R}.$$

(15)

For $u\in Y_j$, with ${\parallel u\parallel }_{X_0}>1$, by (15) and $\left(M_1\right)$, we get

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mu }\left(u\right)& =\tilde{M}\left({\delta }_{p(·)}\left(u\right)\right)-\mu {\int }_{\Omega }\frac{1}{\overline{p}\left(x\right)}{\left|u\right|}^{\overline{p}\left(x\right)}dx-{\int }_{\Omega }G(x,u)dx\hfill \\ \hfill & \le \frac{h_2}{\beta {\left(p^{-}\right)}^{\beta }}{\left({\varrho }_{p(·)}^{s(·)}\left(u\right)\right)}^{\beta }-\frac{{\mu }^{-}}{p^{+}}{\rho }_{\overline{p}(·)}\left(u\right)-c_4{\int }_{\Omega }{\left|u\right|}^{\lambda }dx+c_5{\int }_{\Omega }\left|u\right|dx.\hfill \end{array}$$

On a finite dimensional space $Y_j$ all the norms are equivalent, so there are three positive constants $c_6$, $c_7$, and $c_8$ such that

$${\parallel u\parallel }_{\overline{p}(·)}^{{\overline{p}}^{-}}\ge c_6{\parallel u\parallel }_{X_0}^{{\overline{p}}^{-}}{,\parallel u\parallel }_{\lambda }^{\lambda }\ge c_7{\parallel u\parallel }_{X_0}^{\lambda }{,\parallel u\parallel }_1\ge c_8{\parallel u\parallel }_{X_0}.$$

Consequently, from the above inequalities and Propositions 1 and 2, for any $u\in Y_j$ with ${\parallel u\parallel }_{X_0}>max\left\{1,c_6^{-1/{\overline{p}}^{-}}\right\}$, we have

$${\mathcal{I}}_{\mu }\left(u\right)\le \frac{h_2}{\beta {\left(p^{-}\right)}^{\beta }}{\parallel u\parallel }_{X_0}^{\beta p^{+}}-\frac{{\mu }^{-}}{p^{+}}{c}_6{\parallel u\parallel }_{X_0}^{p^{-}}-c_4{c}_7{\parallel u\parallel }_{X_0}^{\lambda }+c_5{c}_8{\parallel u\parallel }_{X_0}.$$

Since $\lambda >\beta p^{+}>p^{-}>1$ by $\left(G_2\right)$, by choosing ${\rho }_j>max\left\{1,c_6^{-1/{\overline{p}}^{-}}\right\}$ large enough, we get

$$a_j:=\underset{u\in Y_j,{\parallel u\parallel }_{X_0}={\rho }_j}{max}{\mathcal{I}}_{\mu }\left(u\right)\le 0.$$

Therefore, the condition $\left(I_1\right)$ of Theorem 3 holds.

According to $\left(G_1\right)$, (13), and Propositions 1 and 2, we get for any $u\in Z_j$ with ${\parallel u\parallel }_{X_0}>1$

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mu }\left(u\right)& \ge \tilde{M}\left({\delta }_{p(·)}\left(u\right)\right)-{\mu }^{+}{\int }_{\Omega }\frac{1}{\overline{p}\left(x\right)}{\left|u\left(x\right)\right|}^{\overline{p}\left(x\right)}dx-c_1{\int }_{\Omega }\frac{1}{q\left(x\right)}{\left|u\right|}^{q\left(x\right)}dx\hfill \\ \hfill & \ge \frac{h_1}{\beta }{\left({\delta }_{p(·)}\left(u\right)\right)}^{\beta }-\frac{{\mu }^{+}}{p^{-}}{\rho }_{\overline{p}(·)}\left(u\right)-\frac{c_1}{q^{-}}{\rho }_{q(·)}\left(u\right)\hfill \\ \hfill & \ge \frac{h_1}{\beta {\left(p^{+}\right)}^{\beta }}{\parallel u\parallel }_{X_0}^{\beta p^{-}}-\frac{{\mu }^{+}}{p^{-}}max\left\{{\parallel u\parallel }_{\overline{p}(·)}^{{\overline{p}}^{-}},{\parallel u\parallel }_{\overline{p}(·)}^{{\overline{p}}^{+}}\right\}-\frac{c_1}{q^{-}}max\left\{{\parallel u\parallel }_{q(·)}^{q^{-}},{\parallel u\parallel }_{q(·)}^{q^{+}}\right\}\hfill \\ \hfill & \ge \frac{h_1}{\beta {\left(p^{+}\right)}^{\beta }}{\parallel u\parallel }_{X_0}^{\beta p^{-}}-\frac{{\mu }^{+}}{p^{-}}max\left\{({\xi }_j{\parallel u\parallel }_{X_0}{)}^{{\overline{p}}^{-}},({\xi }_j{\parallel u\parallel }_{X_0}{)}^{{\overline{p}}^{+}}\right\}-\frac{c_1}{q^{-}}max\left\{({\xi }_j{\parallel u\parallel }_{X_0}{)}^{q^{-}},({\xi }_j{\parallel u\parallel }_{X_0}{)}^{q^{+}}\right\}\hfill \\ \hfill & \ge \frac{h_1}{\beta {\left(p^{+}\right)}^{\beta }}{\parallel u\parallel }_{X_0}^{\beta p^{-}}-\frac{{\mu }^{+}{\xi }_j^{p^{-}}}{p^{-}}{\parallel u\parallel }_{X_0}^{p^{+}}-\frac{c_1{\xi }_j^{q^{-}}}{q^{-}}{\parallel u\parallel }_{X_0}^{q^{+}}.\hfill \end{array}$$

(16)

We can suppose ${\xi }_j<1$ for j sufficiently large, in view of Lemma 6. Let us define

$${\gamma }_j:={\left(\frac{c_1\beta {\left(q^{-}\right)}^{\beta -1}}{h_1}·{\xi }_j^{q^{-}}\right)}^{\frac{1}{\beta p^{-}-q^{+}}},$$

then since ${\gamma }_j\to +\infty$ as $j\to +\infty$ by Lemma 6 and the fact that $q^{+}\ge q^{-}>\beta p^{+}\ge \beta p^{-}$ by $\left(G_1\right)$, we can assume that ${\gamma }_j>1$ for j larger. Hence, by (16) applied for any $u\in Z_j$ with ${\parallel u\parallel }_{X_0}={\gamma }_j$, we obtain

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mu }\left(u\right)& \ge \frac{h_1}{\beta }\left(\frac{1}{{\left(p^{+}\right)}^{\beta }}-\frac{1}{{\left(q^{-}\right)}^{\beta }}\right){\gamma }_j^{\beta p^{-}}-\frac{{\mu }^{+}{\xi }_j^{p^{-}}}{p^{-}}{\gamma }_j^{p^{+}}\hfill \\ \hfill & ={\gamma }_j^{p^{+}}\left[\frac{h_1}{\beta }\left(\frac{1}{{\left(p^{+}\right)}^{\beta }}-\frac{1}{{\left(q^{-}\right)}^{\beta }}\right){\gamma }_j^{\beta p^{-}-p^{+}}-\frac{{\mu }^{+}{\xi }_j^{p^{-}}}{p^{-}}\right]\to +\infty \hfill \end{array}$$

as $j\to \infty$, by Lemma 6, as also $p^{+}<\beta p^{-}$ by $\left(M_1\right)$ and $q^{-}>\beta p^{+}>p^{+}$ by $\left(G_1\right)$. Thus, the condition $\left(I_2\right)$ of Theorem 3 holds. So for j large enough, $b_j>0$. Theorem 3.5 of [33] implies then the existence of a sequence $\left\{u_n\right\}\subset X_0$ fulfilling

$${\mathcal{I}}_{\mu }\left(u_n\right)\to c_j\mathrm{and}{\mathcal{I}}_{\mu }^{{}^{\prime }}\left(u_n\right)\to 0\mathrm{in}{X}_0^{*}\mathrm{as}n\to \infty .$$

(17)

It follows from the condition $\left(I_3\right)$ that $c_j$ is a critical value of ${\mathcal{I}}_{\mu }$. According to $c_j\ge b_j$ and $b_j\to +\infty ,j\to \infty$, the proof of Theorem 2 is complete, considering that ${\mathcal{I}}_{\mu }$ is even by $\left(G_3\right)$. □

6. Conclusions

In this work, the existence of a solution is obtained by the mountain pass lemma, and the existence of infinitely many solutions with positive energy to Equation (1) is established by using the fountain theorem. We consider a class of complex bi-nonlocal problems, which improves the previous results. In order to overcome the difficulties arising from such problems, we use more sophisticated analytical techniques. This kind of equation has a wide range of applications in many fields, and interested readers may refer to the thin obstacle problem [34,35], ultra-relativistic limits of quantum mechanics [11], finance [12] and so on.