A fractional $p(x,\cdot )$-Laplacian problem involving a singular term

Abstract

This paper deals with a class of singular problems involving the fractional $p(x,\cdot )$-Laplace operator of the form

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p(x,\cdot )}u(x)= \frac{\lambda }{u^{\gamma (x)}}+u^{q(x)-1} &{} \hbox {in }\Omega , \\ u>0, \;\;\text {in}\;\; \Omega &{} \hbox {} \\ u=0 \;\;\text {on}\;\;{\mathbb {R}}^N\setminus \Omega , &{} \hbox {} \end{array} \right. \end{aligned}$$

where $\Omega$ is a smooth bounded domain in ${\mathbb {R}}^N$ ($N\ge 3$), $0<s<1$, $\lambda$ is a positive parameter and $\gamma : {\mathbb {R}}^N \longrightarrow (0,1)$ is a continuous function, $p:\; {\mathbb {R}}^{2N} \longrightarrow \;(1,\infty )$ is a bounded, continuous and symmetric function, $q: {\mathbb {R}}^N \longrightarrow (1,\infty )$ is a continuous function. Using the direct method of minimization combined with the theory of fractional Sobolev spaces with variable exponents, we prove that the problem has one positive solution for $\lambda >0$ small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional $p(x,\cdot )$-Laplace operators.