Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

Canlin Gan, Ting Xiao, Qiongfen Zhang^*

School of Science, Guilin University of Technology, Guilin, China.

DOI: 10.4236/jamp.2022.103057                

Abstract

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R^N, s(⋅) ∈ C (R^N × R^N, (0,1)), (-Δ)^s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

Keywords

Concave-Convex Nonlinearity, Variable-Order Fractional Laplacian, Variational Methods, Fractional Elliptic Equation

1. Introduction

In this paper, we consider the following variable-order fractional Laplacian equations

$\{\begin{array}{ll}{\left(-\Delta \right)}^{s(\cdot )}u+\lambda V\left(x\right)u=f\left(x\mathrm{,}u\right)+\mu {\left|u\right|}^{q\left(x\right)-2}u\mathrm{,}\hfill & \text{in}\Omega \mathrm{,}\hfill \\ u=\mathrm{0,}\hfill & \text{in}{ℝ}^N\backslash \Omega \mathrm{,}\hfill \end{array}$ (1.1)

where $N\ge 1$ and $N>2s\left(x,y\right)$ for $\left(x\mathrm{,}y\right)\in \Omega \times \Omega$, $\Omega$ is a bounded domain in ${ℝ}^N$, $s(\cdot )\in C\left({ℝ}^N\times {ℝ}^N\mathrm{,}\left(\mathrm{0,1}\right)\right)$, ${\left(-\Delta \right)}^{s(\cdot )}$ is the variable-order fractional Laplacian operator, $\lambda ,\mu >0$ are two parameters, $V\mathrm{<mo>\; :\; </mo>}\Omega \to \left[\mathrm{0,}\infty \right)$ is a continuous function, $f\in C\left(\Omega \times ℝ\right)$ and $q\in C\left(\Omega \right)$. The Laplacian operator ${\left(-\Delta \right)}^{s(\cdot )}$ is defined by

${\left(-\Delta \right)}^{s(\cdot )}\xi \left(x\right)=2P\mathrm{.}V\mathrm{.}{\displaystyle {\int }_{{ℝ}^N}}\frac{\xi \left(x\right)-\xi \left(y\right)}{{\left|x-y\right|}^{N+2s\left(x\mathrm{,}y\right)}}\text{d}y\mathrm{,}$

for each $x\in {ℝ}^N$ and any $\xi \in C_0^{\infty }\left(\Omega \right)$, where $P\mathrm{.}V\mathrm{.}$ denotes the Cauchy principal value.

When $s(\cdot )\equiv \text{constant}$ and $q\left(x\right)\equiv \text{constant}$, ${\left(-\Delta \right)}^{s(\cdot )}$ becomes to the usual fractional Laplacian operator and problem (1.1) reduces to fractional Schrödinger equation. This kind of equation is introduced by Laskin [1] [2] as a result of expanding the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. The fractional Schrödinger equation is studied by many researchers. For example, Zhang et al. [3] investigated fractional Schrödinger equation with critical exponents by using variational methods. They used Pohožaev identity and Jeanjean’s monotonicity trick to obtain a radially symmetric weak solution. Another example is [4], the multiplicity and concentration of solutions for fractional Schrödinger equation with concave-convex nonlinearity are studied by Gao et al. For more results about fractional Schrödinger equation, please see [5] [6] [7] and the references therein. Particularly, when $s(\cdot )\equiv \text{1}$ and $q\left(x\right)\equiv \text{constant}$, problem (1.1) becomes to the Schrödinger equation with concave-convex nonlinearity.

If $s(\cdot )\equiv \text{1}$, (1.1) becomes the following second order elliptic equation with variable growth nonlinearity

$\{\begin{array}{ll}-\Delta u+\lambda V\left(x\right)u=f\left(x\mathrm{,}u\right)+\mu {\left|u\right|}^{q\left(x\right)-2}u\hfill & \text{in}\Omega \hfill \\ u=0\hfill & \text{in}{ℝ}^N\backslash \Omega \mathrm{.}\hfill \end{array}$ (1.2)

Some interesting phenomena can be described by this type of model. For example, Ružička [8] showed the application in the modeling of electrorheological fluids involving variable exponent Laplacian operator. It happens that there is a similar case, Ayazoglu and Ekincioglu [9] obtained a positive solution for electrorheological fluids equations with variable exponent via mountain pass technique. For other applications of these similar models, we refer the readers to [10] [11] [12] [13] [14].

In this paper, we consider the variable-order fractional Laplacian operator case with variable growth. The fractional variable order derivatives are introduced by Lorenzo and Hartley [15] to better describe some diffusion processes reacting to temperature changes. In fact, the literature involving the variable-order fractional Laplacian operator cases is few. Specially, Xiang et al. [16] obtained multiple solutions for the following elliptic equations with variable-order fractional Laplacian operator involving variable exponents by using variational methods,

$\{\begin{array}{ll}{\left(-\Delta \right)}^{s(\cdot )}u+\lambda V\left(x\right)u=\alpha {\left|u\right|}^{p\left(x\right)-2}u+\mu {\left|u\right|}^{q\left(x\right)-2}u\hfill & \text{in}\Omega \hfill \\ u=0\hfill & \text{in}{ℝ}^N\backslash \Omega \mathrm{,}\hfill \end{array}$ (1.3)

where $p\mathrm{,}q\in C\left(\Omega \right)$. Another example is [17], Biswas and Tiwari studied a type of Choquard problem with variable-order nonlocal term and variable exponents and obtained some results for the above mention problem by employing Hardy-Sobolev-Littlewood-type inequality. Very recently, Xiang et al. [18] investigated variable-order fractional Kirchhoff equations with nonstandard growth and obtained multiple solutions for these equations by applying the Nehari manifold approach. For other results on variable-order fractional Kirchhoff equations, please see [19] [20] and the references therein.

Inspired mainly by the aforementioned results, we proved the existence of solutions for (1.1) with concave-convex nonlinearity. Compared to [16], we deal with a general case, i.e., the general nonlinearity f with variable growth conditions. To show our result, we make the following assumptions first:

(H1) $0<s^{-}:={\min }_{\left(x,y\right)\in {ℝ}^N\times {ℝ}^N}s\left(x,y\right)\le s^{+}:={\max }_{\left(x,y\right)\in {ℝ}^N\times {ℝ}^N}s\left(x,y\right)<1$.

(H2) $s\left(x\mathrm{,}y\right)=s\left(y\mathrm{,}x\right)$, $\forall \left(x\mathrm{,}y\right)\in {ℝ}^N\times {ℝ}^N$.

(H3) $J=int\left(V^{-1}\left(0\right)\right)\subset \Omega$ is a nonempty bounded domain and $\tilde{J}=V^{-1}\left(0\right)$.

(H4) there exists a nonempty open domain ${\Omega }_0\subset J$ such that $V\left(x\right)\equiv 0$ for all $x\in {\bar{\Omega }}_0$.

For the nonlinearity term f and the variable exponents q, we assume that $q\in C\left(\bar{\Omega }\right)$ and the following assumptions hold:

(H5) $q\mathrm{<mo>\; :\; </mo>}\bar{\Omega }\to \left(\mathrm{1,2}\right)$.

(H6) $f\in C\left(\Omega \times ℝ\right)$ and there exist a positive constant c and a continuous function $p\in C\left(\bar{\Omega }\right)$ with $2<p\left(x\right)<\frac{2N}{N-2s\left(x,x\right)}$ such that $\left|f\left(x\mathrm{,}u\right)\right|\le c\left(1+{\left|u\right|}^{p\left(x\right)-1}\right)$.

(H7) $f\left(x\mathrm{,}u\right)=o\left(\left|u\right|\right)$ as $\left|u\right|\to 0$ uniformly for $x\in \Omega$.

(H8) there exists $p^{-}>2$ such that $0<p^{-}F\left(x,u\right)\le f\left(x,u\right)u$ for every $x\in \Omega$, where $F\left(x\mathrm{,}t\right)={\displaystyle {\int }_0^t}f\left(x\mathrm{,}t\right)\text{d}t$ and

$2<p^{-}:={\text{essinf}}_{x\in \bar{\Omega }}p\left(x\right)\le p^{+}:={\text{esssup}}_{x\in \bar{\Omega }}p\left(x\right)<\frac{2N}{N-2s\left(x,x\right)}$.

Let ${\eta }_p$ and ${\eta }_q$ be the Sobolev embedding constants which will be defined in the next section, set

$A=\frac{\max \left\{{\eta }_p^{p^{+}},{\eta }_p^{p^{-}}\right\}}{p^{-}},B=\frac{\max \left\{{\eta }_q^{q^{+}},{\eta }_q^{q^{-}}\right\}}{q^{-}}.$

We assume that $\mu$ is a positive parameter satisfying the following assumption:

(H9) there holds

$\mu <{\left(\frac{2-q^{+}}{2A{c}_{\epsilon }\left(p^{+}-q^{+}\right)}\right)}^{\frac{2-q^{+}}{p^{+}-2}}\frac{\left(1-{\eta }_2^2\epsilon \right)p^{+}+{\eta }_2^2\epsilon q^{+}-2}{2B\left(p^{+}-q^{+}\right)},c_{\epsilon }\le \frac{2-q^{+}}{2A\left(p^{+}-q^{+}\right)}.$

where $c_{\epsilon }$ is a positive constant depending on $\epsilon$.

Based on the hypothesis (H2), we can give the following definition of weak solutions for problem (1.1).

Definition 1.1. We say that $u\in E_{\lambda }$ is a (weak) solution of problem (1.1), if for any $v\in E_{\lambda }$, there holds

$\begin{array}{l}{\displaystyle {\iint }_{{ℝ}^{2N}}}\frac{\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)}{{\left|x-y\right|}^{N+2s\left(x\mathrm{,}y\right)}}\text{d}x\text{d}y+\lambda {\displaystyle {\int }_{\Omega }}V\left(x\right)uv\text{d}x\\ ={\displaystyle {\int }_{\Omega }}\left(f\left(x\mathrm{,}u\right)v+\mu {\left|u\right|}^{q\left(x\right)-2}uv\right)\text{d}x\mathrm{,}\end{array}$

where $E_{\lambda }$ is a variable exponent Banach space which will be defined in the next section.

Theorem 1.2. Suppose (H1)-(H9) hold. Let $N>2s^{+}$, then problem (1.1) admits at least two distinct solutions for all $\lambda >0$ .

Remark 1.3. In fact, the multiple solutions for variable-order fractional Laplacian equation involving general nonlinearity with critical growth of variable exponent have not been investigated. It is very interesting and full of challenge for us to deal with this problem.

This paper is organized as follows. In Section 1, we give some reviews on the topic of variable-order fractional Lapla-cian equations and give the main result of this paper. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.2.

2. Preliminaries

Some preliminary results of variable exponent Lebesgue spaces will be given in this section which come from [21].

A variable exponent is a measurable function $p\mathrm{<mo>\; :\; </mo>}\bar{\Omega }\to \left[\mathrm{1,}\infty \right)$. The exponent p is said to be bounded if $p^{+}$ is finite. Let

$L^{p\left(x\right)}\left(\Omega \right)=\left\{w\mathrm{<mo>\; :\; </mo>}\Omega \to ℝ\text{is a measurable function;}{\varphi }_{p\left(x\right)}\left(w\right)={\displaystyle {\int }_{\Omega }}{\left|w\left(x\right)\right|}^{p\left(x\right)}\text{d}x<\infty \right\}\mathrm{,}$

then $L^{p\left(x\right)}\left(\Omega \right)$ is a variable exponent Banach space with the following Luxemburg norm

${\Vert w\Vert }_{L^{p(\cdot )}\left(\Omega \right)}=\inf \left\{\theta >\mathrm{0\; <mo>\; :\; </mo>}{\varphi }_{p\left(x\right)}\left({\theta }^{-1}w\right)\le 1\right\}\mathrm{.}$

If p is bounded, there holds

$\min \left\{{\Vert w\Vert }_{L^{p(\cdot )}\left(\Omega \right)}^{p^{-}}\mathrm{,}{\Vert w\Vert }_{L^{p(\cdot )}\left(\Omega \right)}^{p^{+}}\right\}\le {\varphi }_{p(\cdot )}\left(w\right)\le \max \left\{{\Vert w\Vert }_{L^{p(\cdot )}\left(\Omega \right)}^{p^{-}}\mathrm{,}{\Vert w\Vert }_{L^{p(\cdot )}\left(\Omega \right)}^{p^{+}}\right\}\mathrm{.}$ (2.1)

From (2.1), we know that the norm convergence and the convergence is equivalent with respect to ${\varphi }_{p\left(x\right)}$ when p is bounded. Moreover, the dual space of $L^{p(\cdot )}\left(\Omega \right)$ can be written as $L^{p^{\prime }(\cdot )}\left(\Omega \right)$ for bounded exponent, where $1/p^{\prime }\left(x\right)+1/p\left(x\right)=1$. It is obvious that $L^{p(\cdot )}\left(\Omega \right)$ is separable and reflexive since $1<p^{-}\le p^{+}<\infty$.

The following inequality is Hölder’s inequality in variable exponent Lebesgue space

${\displaystyle {\int }_{\Omega }}\left|uv\right|\text{d}x\le \left(\frac{1}{p^{-}}+\frac{1}{{\left(p^{\prime }\right)}^{-}}\right){\Vert u\Vert }_{L^{p(\cdot )}\left(\Omega \right)}{\Vert v\Vert }_{L^{p^{\prime }(\cdot )}\left(\Omega \right)}\le 2{\Vert u\Vert }_{L^{p(\cdot )}\left(\Omega \right)}{\Vert v\Vert }_{L^{p^{\prime }(\cdot )}\left(\Omega \right)}\mathrm{,}$

for all $u\in L^{p(\cdot )}\left(\Omega \right)$, $v\in L^{p^{\prime }(\cdot )}\left(\Omega \right)$ and $p\left(x\right)\in \left(\mathrm{1,}\infty \right)$.

Next, the variational setting for problem (1.1) will be given. Let $\Omega$ be a nonempty open subset of ${ℝ}^N$ and let $s(\cdot )$ be a measurable function, there are two constants $s_0\mathrm{,}{s}_1\in \left(\mathrm{0,1}\right)$ with $s_0<s_1$ such that

$s_0\le s\left(x\mathrm{,}y\right)\le s_1\mathrm{,}\forall \left(x\mathrm{,}y\right)\in {ℝ}^N\times {ℝ}^N\mathrm{.}$

Set

$H^{s(\cdot )}\left(\Omega \right)=\left\{w\in L^2\left(\Omega \right):{\left[w\right]}_{s(\cdot ),\Omega }<\infty \right\},$

where

${\left[w\right]}_{s(\cdot ),\Omega }={\left({\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{\Omega }}\frac{{\left|w\left(x\right)-w\left(y\right)\right|}^2}{{\left|x-y\right|}^{N+2s\left(x\mathrm{,}y\right)}}\text{d}x\text{d}y\right)}^{1/2}\mathrm{.}$

Let $H^{s(\cdot )}\left(\Omega \right)$ be equipped with the norm

${\Vert w\Vert }_{\Omega }={\left({\Vert w\Vert }_{L^2\left(\Omega \right)}^2+{\left[w\right]}_{s(\cdot ),\Omega }^2\right)}^{1/2}\mathrm{.}$

Especially, $H^{s(\cdot )}\left(\Omega \right)$ becomes to the usual fractional Sobolev space if $s(\cdot )$ is a constant function.

The space $H_0^{s(\cdot )}\left(\Omega \right)$ is defined as

$\begin{array}{l}{H}_0^{s(\cdot )}\left(\Omega \right)=\{w\mathrm{<mo>\; :\; </mo>}{ℝ}^N\to ℝ\text{isameasurablefunction}\mathrm{,}{w|}_{{ℝ}^N\backslash \Omega }=\mathrm{0,}{{\displaystyle }}^{}\\ w\in L^2\left(\Omega \right)\text{and}{\left[w\right]}_{s(\cdot )}<\infty \}\mathrm{,}\end{array}$

where

${\left[w\right]}_{s(\cdot )}\mathrm{<mo>\; :\; </mo>}={\left({\displaystyle {\iint }_{{ℝ}^{2N}}}\frac{{\left|w\left(x\right)-w\left(y\right)\right|}^2}{{\left|x-y\right|}^{N+2s\left(x\mathrm{,}y\right)}}\text{d}x\text{d}y\right)}^{1/2}\mathrm{.}$

The norm on $H_0^{s(\cdot )}\left(\Omega \right)$ is given as

$\Vert w\Vert ={\left({\Vert w\Vert }_{L^2\left(\Omega \right)}^2+{\left[w\right]}_{s(\cdot )}^2\right)}^{1/2}\mathrm{.}$

The following lemma implies that ${[\cdot ]}_{s(\cdot )}$ is an equivalent norm of $H_0^{s(\cdot )}\left(\Omega \right)$.

Lemma 2.1. [16]. The embedding $H_0^{s_1}\left(\Omega \right)$ ↪ $H_0^{s(\cdot )}\left(\Omega \right)$ ↪ $H_0^{s_0}\left(\Omega \right)$ are continuous. Moreover, if $N>2s_0$, for any fixed constant exponent

$p\in \left[\mathrm{1,}\frac{2N}{N-2s_0}\right]$, $H_0^{s(\cdot )}\left(\Omega \right)$ can be continuously embedded into $L^p\left(\Omega \right)$.

Using the same argument as the proof of ( [22], Lemma 7), one can easily prove that $\left(H_0^{s(\cdot )}\left(\Omega \right)\mathrm{,}{[\cdot ]}_{s(\cdot )}\right)$ is a Hilbert space. From ( [22], Theorem 2.1), we know that the embedding $H_0^{s(\cdot )}\left(\Omega \right)$ ↪ $L^{p\left(x\right)}\left(\Omega \right)$ is continuous and compact. Moreover, there exists ${\eta }_p=\eta \left(N,s^{+},s^{-},p^{+}\right)>0$ such that

${\Vert w\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\le {\eta }_p\Vert u\Vert \mathrm{,}\forall w\in H_0^{s(\cdot )}\left(\Omega \right)\mathrm{,}$ (2.2)

and

${\Vert w\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\le {\eta }_p{\left[w\right]}_{s(\cdot )}\mathrm{,}\forall w\in H_0^{s(\cdot )}\left(\Omega \right)\mathrm{.}$ (2.3)

Let

$E=\left\{w\in H_0^{s(\cdot )}\left(\Omega \right):{\left[w\right]}_{s(\cdot )}^2+{\displaystyle {\int }_{\Omega }}V\left(x\right){\left|w\right|}^2\text{d}x<\infty \right\},$

the inner product on E is defined as

$\begin{array}{l}{\left(v,w\right)}_E={\displaystyle {\iint }_{{ℝ}^{2N}}}\frac{\left(v\left(x\right)-v\left(y\right)\right)\left(w\left(x\right)-w\left(y\right)\right)}{{\left|x-y\right|}^{N+2s\left(x\mathrm{,}y\right)}}\text{d}x\text{d}y\\ +{\displaystyle {\int }_{\Omega }}V\left(x\right)vw\text{d}x\forall v,w\in E,\end{array}$

and the corresponding norm is ${\Vert w\Vert }_E={\left(w\mathrm{,}w\right)}_E^{1/2}$. The following inner product

${\left(v\mathrm{,}w\right)}_{\lambda }={\displaystyle {\iint }_{{ℝ}^{2N}}}\frac{\left(v\left(x\right)-v\left(y\right)\right)\left(w\left(x\right)-w\left(y\right)\right)}{{\left|x-y\right|}^{N+2s\left(x\mathrm{,}y\right)}}\text{d}x\text{d}y+\lambda {\displaystyle {\int }_{\Omega }}V\left(x\right)vw\text{d}x\mathrm{,}$

and the corresponding norm ${\Vert w\Vert }_{\lambda }={\left(w\mathrm{,}w\right)}_{\lambda }^{1/2}$ are also used in this paper. Obviously, for all $\lambda \ge 1$, one has ${\Vert w\Vert }_E\le {\Vert u\Vert }_{\lambda }$. Set $E_{\lambda }=\left(E\mathrm{,}{\Vert \cdot \Vert }_{\lambda }\right)$. Moreover, for $p\left(x\right)\in \left(\mathrm{1,}2N/\left(N-2s\left(x\mathrm{,}x\right)\right)\right)$, from (2.3), one has

${\displaystyle {\int }_{\Omega }}{\left|w\left(x\right)\right|}^{p\left(x\right)}\text{d}x\le \max \left\{{\Vert w\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}^{p^{+}}\mathrm{,}{\Vert w\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}^{p^{-}}\right\}\le \max \left\{{\eta }_p^{p^{+}}{\left[w\right]}_{s(\cdot )}^{p^{+}}\mathrm{,}{\eta }_p^{p^{-}}{\left[w\right]}_{s(\cdot )}^{p^{-}}\right\}\mathrm{.}$

Evidently, problem (1.1) has a variational formulation and the corresponding functional is defined in $E_{\lambda }$ by

$\begin{array}{c}{\Phi }_{\lambda }\left(u\right)=\frac{1}{2}{\displaystyle {\iint }_{{ℝ}^{2N}}}\frac{{\left|u\left(x\right)-u\left(y\right)\right|}^2}{{\left|x-y\right|}^{N+2s\left(x\mathrm{,}y\right)}}\text{d}x\text{d}y+\frac{\lambda }{2}{\displaystyle {\int }_{\Omega }}V\left(x\right){\left|u\right|}^2\text{d}x\\ -{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}u\right)\text{d}x-\frac{\mu }{q\left(x\right)}{\displaystyle {\int }_{\Omega }}{\left|u\right|}^{q\left(x\right)}\text{d}x\\ =\frac{1}{2}{\Vert u\Vert }_{\lambda }^2-{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}u\right)\text{d}x-\frac{\mu }{q\left(x\right)}{\displaystyle {\int }_{\Omega }}{\left|u\right|}^{q\left(x\right)}\text{d}x\mathrm{,}\forall u\in E_{\lambda }\mathrm{.}\end{array}$ (2.4)

Actually, ${\Phi }_{\lambda }\in C^1\left(E_{\lambda }\mathrm{,}ℝ\right)$ is well-defined and

$\langle {\Phi }^{\prime }\left(u\right)\mathrm{,}v\rangle ={\left(u\mathrm{,}v\right)}_{\lambda }-\mu {\displaystyle {\int }_{\Omega }}{\left|u\right|}^{q\left(x\right)-2}uv\text{d}x-{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}u\right)v\text{d}x\mathrm{,}\forall u\mathrm{,}v\in E_{\lambda }\mathrm{.}$ (2.5)

Hence, u is a solution to problem (1.1) if $u\in E_{\lambda }$ is a critical point of ${\Phi }_{\lambda }$.

3. Proof of Theorem 1.2

It is well known that a $C^1$ functional ${\Phi }_{\lambda }$ satisfying Palais-Smale ((PS) for short) condition at level c if for any sequence $\left\{u_n\right\}\subset E_{\lambda }$ such that ${\Phi }_{\lambda }\left(u_n\right)\to c$ and ${{\Phi }^{\prime }}_{\lambda }\left(u_n\right)\to 0$, there exists a convergent subsequence in $E_{\lambda }$, which is called a (PS)_c sequence.

First, we shall verify the mountain pass geometry of ${\Phi }_{\lambda }$.

Lemma 3.1. Assume that (H1) and (H3)-(H9) hold. Then for all $\lambda >0$, the functional ${\Phi }_{\lambda }$ satisfies

1) There exists $\beta ,\rho >0$ such that ${\Phi }_{\lambda }\left(u\right)\ge \beta$ if ${\Vert u\Vert }_{\lambda }=\rho$ ;

2) There exists $u\in E_{\lambda }$ with ${\Vert u\Vert }_{\lambda }>\rho$ such that ${\Phi }_{\lambda }\left(u\right)<0$.

Proof. For any $\epsilon >0$, it follows from the condition (H5) and (H6) that there exists $c_{\epsilon }>0$ such that

$\left|F\left(x\mathrm{,}t\right)\right|\le \frac{\epsilon }{2}{\left|t\right|}^2+\frac{c_{\epsilon }}{p^{-}}{\left|t\right|}^{p\left(x\right)}\mathrm{,}\forall t\in ℝ\mathrm{.}$ (3.1)

Thus, from (3.1) and the fractional Sobolev inequality, one has

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}u\right)\text{d}x\le \frac{\epsilon }{2}{\displaystyle {\int }_{\Omega }}{\left|u\right|}^2\text{d}x+\frac{c_{\epsilon }}{p^{-}}{\displaystyle {\int }_{\Omega }}{\left|u\right|}^{p\left(x\right)}\text{d}x\\ \le \frac{C_2^2\epsilon }{2}{\Vert u\Vert }_{\lambda }^2+\frac{c_{\epsilon }\max \left\{{\eta }_p^{p^{+}}\mathrm{,}{\eta }_p^{p^{-}}\right\}}{p^{-}}{\Vert u\Vert }_{\lambda }^{p^{+}}\mathrm{,}\forall u\in E_{\lambda }\text{with}{\Vert u\Vert }_{\lambda }\ge 1.\end{array}$ (3.2)

For $u\in E_{\lambda }$ and ${\Vert u\Vert }_{\lambda }\ge 1$, it follows from (2.4) and (3.2) that

$\begin{array}{l}{\Phi }_{\lambda }\left(u\right)=\frac{1}{2}{\Vert u\Vert }_{\lambda }^2-{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}u\right)\text{d}x-\frac{\mu }{q\left(x\right)}{\displaystyle {\int }_{\Omega }}{\left|u\right|}^{q\left(x\right)}\text{d}x\\ \ge \frac{1}{2}{\Vert u\Vert }_{\lambda }^2-\frac{C_2^2\epsilon }{2}{\Vert u\Vert }_{\lambda }^2-\frac{c_{\epsilon }\max \left\{{\eta }_p^{p^{+}}\mathrm{,}{\eta }_p^{p^{-}}\right\}}{p^{-}}{\Vert u\Vert }_{\lambda }^{p^{+}}-\frac{\mu \max \left\{{\eta }_q^{q^{+}}\mathrm{,}{\eta }_q^{q^{-}}\right\}}{q^{-}}{\Vert u\Vert }_{\lambda }^{q^{+}}\\ ={\Vert u\Vert }_{\lambda }^{q^{+}}\left[\frac{1}{2}\left(1-C_2^2\epsilon \right){\Vert u\Vert }_{\lambda }^{2-q^{+}}-\frac{c_{\epsilon }\max \left\{{\eta }_p^{p^{+}}\mathrm{,}{\eta }_p^{p^{-}}\right\}}{p^{-}}{\Vert u\Vert }_{\lambda }^{p^{+}-q^{+}}-\frac{\mu \max \left\{{\eta }_q^{q^{+}}\mathrm{,}{\eta }_q^{q^{-}}\right\}}{q^{-}}\right]\mathrm{.}\end{array}$

Let

$g\left(t\right)=h\left(t\right)t^{q^{+}}\mathrm{,}\forall t\ge \mathrm{0,}$

where

$h\left(t\right)=\frac{1-{\eta }_2^2\epsilon }{2}{t}^{2-q^{+}}-A{c}_{\epsilon }{t}^{p^{+}-q^{+}}-B\mu \mathrm{.}$

By (H9) and an easy computation, for $t=t^{*}={\left[\frac{2-q^{+}}{2A{c}_{\epsilon }\left(p^{+}-q^{+}\right)}\right]}^{\frac{1}{p^{+}-2}}$, one has

$h\left(t^{*}\right)=\underset{t\ge 0}{\max }h\left(t\right)>0.$

Since $c_{\epsilon }\le \frac{2-q^{+}}{2A\left(p^{+}-q^{+}\right)}$, we have

$t^{*}={\left[\frac{2-q^{+}}{2A{c}_{\epsilon }\left(p^{+}-q^{+}\right)}\right]}^{\frac{1}{p^{+}-2}}\ge 1.$

Let $\rho =t^{\mathrm{<mo>\; *\; </mo>}}$ and $\beta =g\left(t^{\mathrm{<mo>\; *\; </mo>}}\right)$, then (1) of Lemma 3.1 is satisfied by.

By (3.1) and (H7), there exists $c_1>0$ such that

$F\left(x,u\right)\ge c_1\left({\left|u\right|}^{p^{-}}-{\left|u\right|}^2\right),\forall \left(x,u\right)\in \Omega \times ℝ.$

Then we choose a function $v_0\in E_{\lambda }$ such that

${\Vert v_0\Vert }_{\lambda }=1\text{and}{\displaystyle {\int }_{\Omega }}{\left|v_0\left(x\right)\right|}^{p\left(x\right)}\text{d}x>0.$

By (2.4), for all $t_0\ge 1$, we obtain

$\begin{array}{c}{\Phi }_{\lambda }\left(t_0{v}_0\right)=\frac{t_0^2}{2}{\Vert v_0\Vert }_{\lambda }^2-{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}{t}_0{v}_0\right)\text{d}x-\frac{\mu }{q\left(x\right)}{\displaystyle {\int }_{\Omega }}{\left|t_0{v}_0\right|}^{q\left(x\right)}\text{d}x\\ \le \frac{t_0^2}{2}{\Vert v_0\Vert }_{\lambda }^2-c_1{t}_0^{p^{-}}{\displaystyle {\int }_{\Omega }}{\left|v_0\right|}^{p\left(x\right)}\text{d}x+c_1{t}_0^2{\displaystyle {\int }_{\Omega }}{\left|v_0\right|}^2\text{d}x-\frac{\mu t_0^{q^{-}}}{q^{+}}{\displaystyle {\int }_{\Omega }}{\left|v_0\right|}^{q\left(x\right)}\text{d}x\mathrm{.}\end{array}$

Since $p^{-}>2$ and $q^{-}>1$, there exists $t_0\ge 1$ large enough such that ${\Phi }_{\lambda }\left(t_0{v}_0\right)<0$. The proof is completed.

Let

$c_{\lambda }=\underset{\rho \in \Gamma }{\inf }\underset{t\in \left[\mathrm{0,1}\right]}{\max }{\Phi }_{\lambda }\left(\rho \left(t\right)\right)\mathrm{,}$

$c\left({\Omega }_0\right)=\underset{\rho \in \tilde{\Gamma }}{\inf }\underset{t\in \left[\mathrm{0,1}\right]}{\max }{{\Phi }_{\lambda }|}_{H_0^{s(\cdot )}\left({\Omega }_0\right)}\left(\rho \left(t\right)\right)\mathrm{,}$

$\Gamma =\left\{\rho \in C\left(\left[0,1\right],E_{\lambda }\right);\rho \left(0\right)=0,\rho \left(1\right)=e\right\}$

and

$\tilde{\Gamma }=\left\{\rho \in C\left(\left[0,1\right],H_0^{s(\cdot )}\left({\Omega }_0\right)\right);\rho \left(0\right)=0,\rho \left(1\right)=e\right\}.$

Notice that

${{\Phi }_{\lambda }|}_{H_0^{s(\cdot )}\left({\Omega }_0\right)}\left(u\right)=\frac{1}{2}{\left[u\right]}_{s(\cdot )}^2-{\displaystyle {\int }_{{\Omega }_0}}\left(F\left(x\mathrm{,}u\right)+\frac{\mu }{q\left(x\right)}{\left|u\right|}^{q\left(x\right)}\right)\text{d}x\mathrm{,}\forall u\in H_0^{s(\cdot )}\left({\Omega }_0\right)\mathrm{.}$

Obviously, $c\left({\Omega }_0\right)$ is independent of $\lambda$. It is clear that the mountain pass geometry of ${{\Phi }_{\lambda }|}_{H_0^{s(\cdot )}\left({\Omega }_0\right)}$ is proved by Lemma 3.1. Since $H_0^{s(\cdot )}\left({\Omega }_0\right)\subset E_{\lambda }$ for all $\lambda >0$, we have $0\le c_{\lambda }\le c\left({\Omega }_0\right)$ for all $\lambda >0$. Evidently, for all $t\in \left[\mathrm{0,1}\right]$, $te\in \tilde{\Gamma }$. Hence, it follows from $p^{-}>2$ that there exists $C_0>0$ such that

$c\left({\Omega }_0\right)\le \underset{t\in \left[0,1\right]}{\max }{\Phi }_{\lambda }\left(te\right)\le C_0<\infty .$

Then, for all $\lambda >0$, we have

$0<\beta \le c_{\lambda }\le c\left({\Omega }_0\right)<C_0.$ (3.3)

By Lemma 3.1 and the mountain pass theorem, we derive that there exists ${\left\{u_k\right\}}_k\subset E_{\lambda }$ such that

${\Phi }_{\lambda }\left(u_k\right)\to c_{\lambda }>0\text{and}{{\Phi }^{\prime }}_{\lambda }\left(u_k\right)\to \mathrm{0,}\text{as}k\to \infty \mathrm{.}$ (3.4)

Lemma 3.2. Assume that (H1) and (H3)-(H9) hold. Then the (PS)_c sequence ${\left\{u_k\right\}}_k\subset E_{\lambda }$ given in (3.4) is bounded for all $\lambda >0$ .

Proof. By (2.4), (2.5), (3.4) (H7) and the Hölder inequality, one obtains

$\begin{array}{c}{c}_{\lambda }+o\left(1\right)\ge {\Phi }_{\lambda }\left(u_k\right)-\frac{1}{p^{-}}\langle {{\Phi }^{\prime }}_{\lambda }\left(u_k\right)\mathrm{,}{u}_k\rangle \\ =\left(\frac{1}{2}-\frac{1}{p^{-}}\right){\Vert u_k\Vert }_{\lambda }^2+{\displaystyle {\int }_{\Omega }}\left[\frac{1}{p^{-}}f\left(x\mathrm{,}{u}_k\right)u_k-F\left(x\mathrm{,}{u}_k\right)\right]\text{d}x\\ -{\displaystyle {\int }_{\Omega }}\mu \left(\frac{1}{q\left(x\right)}-\frac{1}{p^{-}}\right){\left|u_k\right|}^{q\left(x\right)}\text{d}x\\ \ge \frac{p^{-}-2}{2p^{-}}{\Vert u_k\Vert }_{\lambda }^2-\left(\frac{1}{q^{-}}-\frac{1}{p^{-}}\right){\displaystyle {\int }_{\Omega }}\mu {\left|u_k\right|}^{q\left(x\right)}\text{d}x\\ \ge \frac{p^{-}-2}{2p^{-}}{\Vert u_k\Vert }_{\lambda }^2-\mu \left(\frac{1}{q^{-}}-\frac{1}{p^{-}}\right)\max \left\{C_q^{q^{+}}{\Vert u_k\Vert }^{q^{+}}\mathrm{,}{C}_q^{q^{-}}{\Vert u_k\Vert }^{q^{-}}\right\}\mathrm{.}\end{array}$ (3.5)

Arguing indirectly, we assume that ${\left\{u_k\right\}}_k$ is not bounded in $H_0^{s(\cdot )}\left(\Omega \right)$. Then there exists a subsequence still denoted by ${\left\{u_k\right\}}_k$ such that ${\Vert u_k\Vert }_{\lambda }\to \infty$ as $k\to \infty$. Then, by (3.5), we obtain

$\frac{c_{\lambda }+o\left(1\right)}{{\Vert u_k\Vert }_{\lambda }^2}\ge \frac{p^{-}-2}{2p^{-}}-\max \left\{C_q^{q^{+}}{\Vert u_k\Vert }^{q^{+}-2}\mathrm{,}{C}_q^{q^{-}}{\Vert u_k\Vert }^{q^{-}-2}\right\}\mathrm{,}$ (3.6)

which contradicts to $p^{-}>2$. Hence, the boundedness of ${\left\{u_k\right\}}_k$ in $E_{\lambda }$ is obtained for all $\lambda >0$.

Lemma 3.3. Assume that (H1) and (H3)-(H9) hold. Then ${\Phi }_{\lambda }$ satisfies the (PS)_c condition in $E_{\lambda }$ for all $c\in ℝ$ and $\lambda >0$ .

Proof. Let ${\left\{u_k\right\}}_k$ be a (PS)_c sequence with $c<C_0$. From Lemma 3.2, we know that ${\left\{u_k\right\}}_k$ is bounded in $E_{\lambda }$ and there exists $C>0$ such that ${\Vert u_k\Vert }_{\lambda }\le C$. Hence, there exists a subsequence of ${\left\{u_k\right\}}_k$ still denoted by ${\left\{u_k\right\}}_k$ and $u_0$ in $E_{\lambda }$ such that

$\begin{array}{l}{u}_k\rightharpoonup u_0\text{weaklyin}{E}_{\lambda }\mathrm{,}\\ u_k\to u_0\text{a}\text{.e}\text{.in}ℝ\mathrm{,}\\ {\left|u_k\right|}^{p\left(x\right)-2}{u}_k\rightharpoonup {\left|u_0\right|}^{p\left(x\right)-2}{u}_0\text{weaklyin}{L}^{p^{\prime }\left(x\right)}\left(\Omega \right)\mathrm{.}\end{array}$ (3.7)

Now we prove that $u_k\to u_0$ in $E_{\lambda }$. From ( [22], Theorem 2.1), we have $u_k\to u_0$ in $L^{p\left(x\right)}\left(\Omega \right)$ and $L^{q\left(x\right)}\left(\Omega \right)$, respectively. Hence,

$\underset{k\to \infty }{\lim }{\displaystyle {\int }_{\Omega }}{\left|u_k-u_0\right|}^{q\left(x\right)}\text{d}x=0.$ (3.8)

By (H5) and (H6), we have

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_k-u_0\right)\left(u_k-u_0\right)\text{d}x\\ \le {\displaystyle {\int }_{\Omega }}\left(\epsilon \left|u_k-u_0\right|+c_{\epsilon }{\left|u_k-u_0\right|}^{p\left(x\right)-1}\right)\left(u_k-u_0\right)\text{d}x\\ ={\displaystyle {\int }_{\Omega }}\left(\epsilon {\left|u_k-u_0\right|}^2+c_{\epsilon }{\left|u_k-u_0\right|}^{p\left(x\right)}\right)\text{d}x\end{array}$

Thus,

$\underset{k\to \infty }{\lim }{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_k-u_0\right)\left(u_k-u_0\right)\text{d}x=0.$ (3.9)

It follows from (2.5) and (3.4) that

$\begin{array}{c}o\left(1\right)=\langle {{\Phi }^{\prime }}_{\lambda }\left(u_k\right)-{{\Phi }^{\prime }}_{\lambda }\left(u_0\right)\mathrm{,}{u}_k-u_0\rangle \\ ={\left(u_k-u_0\mathrm{,}{u}_k-u_0\right)}_{\lambda }-{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_k-u_0\right)\left(u_k-u_0\right)\text{d}x\\ -\mu {\displaystyle {\int }_{\Omega }}\left({\left|u_k\right|}^{q\left(x\right)-2}{u}_k-{\left|u_0\right|}^{q\left(x\right)-2}{u}_0\right)\left(u_k-u_0\right)\text{d}x\mathrm{.}\end{array}$ (3.10)

By (3.8), (3.9) and (3.10), we have

$\underset{k\to \infty }{\lim }{\Vert u_k-u_0\Vert }_{\lambda }=0.$

The proof is complete.

Proof of Theorem 1.2. By Lemmas 3.1 - 3.2 and the mountain pass theorem, for all $\lambda >0$, there exists a ${\left(PS\right)}_{c_{\lambda }}$ sequence ${\left\{u_k\right\}}_k$ for ${\Phi }_{\lambda }$ on $E_{\lambda }$. From Lemma 3.2 and $0<c_{\lambda }<c\left({\Omega }_0\right)<C_0$, there exists a subsequence of ${\left\{u_k\right\}}_k$ still denoted by ${\left\{u_k\right\}}_k$ and $u_{\lambda }^0\in E_{\lambda }$ such that $u_k\to u_{\lambda }^0$ in $E_{\lambda }$. Moreover, ${\Phi }_{\lambda }\left(u_n\right)=c_{\lambda }\ge \beta$ and $u_{\lambda }^0$ is a solution to problem (1.1).

Next, we verify that problem (1.1) has another solution. Let

${\tilde{c}}_{\lambda }=\inf \left\{{\Phi }_{\lambda }\left(w\right)\mathrm{<mo>\; :\; </mo>}w\in \overline{B_{\rho }}\right\}\mathrm{,}$

where $B_{\rho }=\left\{w\in E_{\lambda }:{\Vert w\Vert }_{\lambda }<\rho \right\}$ and $\rho >0$ is given by Lemma 3.1. Then ${\tilde{c}}_{\lambda }<0$ for all $\lambda >0$. Let $w_0\in H_0^{s(\cdot )}\left(\Omega \right)$ such that ${\displaystyle {\int }_{\Omega }}{\left|w_0\right|}^{q\left(x\right)}>0$. By (2.4), (H7) and (H8), one has that

$\begin{array}{c}{\Phi }_{\lambda }\left(t{w}_0\right)=\frac{t^2}{2}{\Vert w_0\Vert }_{\lambda }^2-{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}t{w}_0\right)\text{d}x-\frac{\mu }{q\left(x\right)}{\displaystyle {\int }_{\Omega }}{\left|t{w}_0\right|}^{q\left(x\right)}\text{d}x\\ \le \frac{t^2}{2}{\Vert w_0\Vert }_{\lambda }^2-c_1{t}^{p^{-}}{\displaystyle {\int }_{\Omega }}{\left|w_0\right|}^{p\left(x\right)}\text{d}x+c_1{t}^2{\displaystyle {\int }_{\Omega }}{\left|w_0\right|}^2\text{d}x-\frac{\mu t_0^{q^{-}}}{q^{+}}{\displaystyle {\int }_{\Omega }}{\left|w_0\right|}^{q\left(x\right)}\text{d}x\\ <0\end{array}$ (3.11)

for $t\in \left(\mathrm{0,1}\right)$ small enough.

Hence, there is $w_0\in E_{\lambda }$ such that ${\Phi }_{\lambda }\left(t{w}_0\right)<0$ for all $t>0$ small enough.

It follows from Lemma 3.1 and the Ekeland’s variational principle that there exists a sequence ${\left\{u_k\right\}}_k$ such that

${\tilde{c}}_{\lambda }\le {\Phi }_{\lambda }\left(u_k\right)\le {\tilde{c}}_{\lambda }+\frac{1}{k}\mathrm{,}$ (3.12)

and

${\Phi }_{\lambda }\left(v\right)\ge {\Phi }_{\lambda }\left(u_k\right)-\frac{{\Vert u_k-v\Vert }_{\lambda }}{k}\mathrm{,}\forall v\in \overline{B_{\rho }}\mathrm{.}$ (3.13)

Now we should show that ${\Vert u_k\Vert }_{\lambda }<\rho$ for k large enough. Indirectly, we suppose that ${\Vert u_k\Vert }_{\lambda }=\rho$ for infinitely many k. Without loss of generality, one may assume that ${\Vert u_k\Vert }_{\lambda }=\rho$ for any $k\in \mathbb{N}$. By Lemma 3.1, one has

${\Phi }_{\lambda }\left(u_k\right)\ge \beta >0.$ (3.14)

From (3.12) and (3.14), we have ${\tilde{c}}_{\lambda }\ge \beta >0$, which contradicts to ${\tilde{c}}_{\lambda }<0$. Now we prove that ${{\Phi }^{\prime }}_{\lambda }\left(u_k\right)\to 0$ in $E_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}$. Let

$w_k=u_k+tv\mathrm{,}\forall v\in B_1\mathrm{,}$

where $B_1:=\left\{v\in E_{\lambda }:{\Vert v\Vert }_{\lambda }=1\right\}$, $t>0$ is small enough such that ${\rho }^2-{\Vert u_k\Vert }_{\lambda }^2-2t\rho -t^2\ge 0$ for fixed k large. Thus

${\Vert w_k\Vert }_{\lambda }^2={\Vert u_k\Vert }_{\lambda }^2+2t\rho {\langle u_k\mathrm{,}v\rangle }_{\lambda }+t^2\le {\Vert u_k\Vert }_{\lambda }^2+2\rho t+t^2\le {\rho }^2\mathrm{,}$

which implies that $w_k\in \overline{B_{\rho }}$. Hence, from (3.13), one has

${\Phi }_{\lambda }\left(w_k\right)\ge {\Phi }_{\lambda }\left(u_k\right)-\frac{t}{n}\Vert u_k-w_k\Vert \mathrm{,}$

that is,

$\frac{{\Phi }_{\lambda }\left(u_k+tv\right)-{\Phi }_{\lambda }\left(u_k\right)}{t}\ge -\frac{1}{k}\mathrm{.}$

Letting $t\to 0^{+}$, one has $\langle {{\Phi }^{\prime }}_{\lambda }\left(u_k\right)\mathrm{,}v\rangle \ge -\frac{1}{k}$ for any fixed k large. Similarly, by repeating the process above, choosing $t<0$ and $\left|t\right|$ small enough, one gets

$\langle {{\Phi }^{\prime }}_{\lambda }\left(u_k\right)\mathrm{,}v\rangle \le \frac{1}{k}\mathrm{,}\text{foranyfixed}k\text{large}\mathrm{.}$

Hence, we obtain

$\underset{n\to \infty }{\lim }\underset{v\in B_1}{\sup }\left|\langle {\Phi }_{\lambda }\left(u_k\right)\mathrm{,}v\rangle \right|=\mathrm{0,}$

which implies that ${\Phi }_{\lambda }\left(u_k\right)\to 0$ in $E_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}$ as $k\to \infty$. Thus, ${\left\{u_n\right\}}_n$ is a ${\left(PS\right)}_{{\tilde{c}}_{\lambda }}$ sequence for the functional ${\Phi }_{\lambda }$. Using the same argument as Lemma 3.3, there exists $u_{\lambda }^{\left(2\right)}$ such that $u_n\to u_{\lambda }^{\left(2\right)}$ in $E_{\lambda }$. Hence, one obtains a nontrivial solution $u_{\lambda }^{\left(2\right)}$ of (1.1) satisfying

${\Phi }_{\lambda }\left(u_{\lambda }^{\left(2\right)}\right)\le \zeta <0\text{and}{\Vert u_{\lambda }^{\left(2\right)}\Vert }_{\lambda }<\rho \mathrm{.}$

Hence, we conclude that

${\Phi }_{\lambda }\left(u_{\lambda }^{\left(2\right)}\right)={\tilde{c}}_{\lambda }\le \zeta <0<\beta <c_{\lambda }={\Phi }_{\lambda }\left(u_{\lambda }^{\left(1\right)}\right),\forall \lambda >0,$

which completes the proof.

4. Conclusion

This paper considers the existence of solutions for a kind of variable-order fractional Laplacian equations. By employing the mountain pass theorem and Ekeland’s variational principle, two solutions are obtained under some suitable conditions on f. Specially speaking, we first prove the mountain pass geometry of the function for this kind of variable-order fractional Laplacian equations. Secondly, we verify the boundedness of (PS)_c sequences. Finally, we prove that ${\Phi }_{\lambda }$ satisfies the (PS)_c condition in $E_{\lambda }$ for all $c\in ℝ$ and $\lambda >0$. The result obtained in this paper generalizes the related ones in the literature, which can be used in similar kinds of variable-order fractional Laplacian equations. We hope that this result can be widely used in variable-order fractional Laplacian equations and other systems.

Funding

This work is supported by the National Natural Science Foundation of China (No. 11961014) and Guangxi Natural Science Foundation (2021GXNSFAA196040).

Authors’ Contributions

The authors have the same contributions in writing this paper.

Conflicts of Interest

All authors declare no conflicts of interest in this paper.

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