Negative Energy Solutions for a New Fractional p x -Kirchhoff Problem without the (AR) Condition

Abstract

In this paper, we investigate the following Kirchhoff type problem involving the fractional $p\left(x\right)$ -Laplacian operator. $\left\{\begin{array}{c}\left(a-b{\int }_{\Omega \times \Omega }\left({\left|u\left(x\right)-u\left(y\right)\right|}^{p\left(x,y\right)}/p\left(x,y\right){\left|x-y\right|}^{N+sp\left(x,y\right)}\right)dxdy\right)\mathcal{L}u=\lambda {\left|u\right|}^{q\left(x\right)-2}u+f\left(x,u\right)\text{x}\in \Omega \hfill \\ u=0\text{x}\in \partial \Omega ,\hfill \end{array}\right.$ , where $\Omega$ is a bounded domain in ${ℝ}^N$ with Lipschitz boundary, $a\ge b>0$ are constants, $p\left(x,y\right)$ is a function defined on $\overline{\Omega }\times \overline{\Omega }$ , $s\in \left(0,1\right)$ , and $q\left(x\right)>1$ , $\mathcal{L}u$ is the fractional $p\left(x\right)$ -Laplacian operator, $N>sp\left(x,y\right)$ , for any $\left(x,y\right)\in \overline{\Omega }\times \overline{\Omega }$ , $p{\left(x\right)}^{\ast }=\left(p\left(x,x\right)N\right)/\left(N-sp\left(x,x\right)\right)$ , $\lambda$ is a given positive parameter, and $f$ is a continuous function. By using Ekeland’s variational principle and dual fountain theorem, we obtain some new existence and multiplicity of negative energy solutions for the above problem without the Ambrosetti-Rabinowitz ((AR) for short) condition.