Existence results for a Kirchhoff-type equation involving fractional p(x)-Laplacian

Abstract

The purpose of this paper is to investigate the existence of weak solutions for a Kirchhoff-type problem driven by a non-local integro-differential operator as follows: $ \begin{equation*} \left\{ \begin{aligned} &M\left(\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)|x-y|^{N+sp(x,y)}}dxdy\right)(-\Delta_{p(x)})^{s}u(x) = f(x,u)&{\rm{in}}\,\,&\Omega,\\ &\quad u = 0 &{\rm{in}}\,\,&\mathbb{R}^{N}\backslash\Omega,\\ \end{aligned}\right. \end{equation*} $ where $ \Omega $ is a smooth bounded open set in $ \mathbb{R}^N $, $ s\in (0, 1) $ and $ p $ is a positive continuous function with $ sp(x, y) < N $, $ M $ and $ f $ are two continuous functions, $ (-\Delta_{p(x)})^{s} $ is the fractional $ p(x) $-Laplacian operator. Using variational methods combined with the theory of the generalized Lebesgue Sobolev space, we prove the existence of nontrivial solution for the problem in an appropriate space of functions.

Keywords

• fractional p(x)-laplace operator
• kirchhoff-type equation
• variational methods