On some singular problems involving the fractional p(x,.) -Laplace operator

Abstract

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular problem involving the fractional $p(x,.)$-Laplace operator $\{\begin{array}{l}(-\mathrm{\Delta }{)}_{p(x,.)}^{s}u+|u{|}^{q(x)-2}u=g(x)u^{ϵ-1-\gamma (x)}\mp \lambda f(x,u)\text{in }\mathrm{\Omega },\\ \\ u=0\text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$where Ω is a smooth bounded domain in ${\mathbb{R}}^N$ ($N\ge 3$), $0<s,ϵ<1$, λ is a positive parameter and $\gamma :\overline{\mathrm{\Omega }}⟶(0,ϵ)$ is a continuous function, $p:\overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}⟶(1,\mathrm{\infty })$ is a bounded, continuous and symmetric function, $q:\overline{\mathrm{\Omega }}⟶(1,\mathrm{\infty })$ is a continuous function, $g\in L^{\frac{p_s^{\ast }(x)-ϵ}{p_s^{\ast }(x)+\gamma (x)-2ϵ}}(\mathrm{\Omega })$ and $g(x)>0$ with $p_s^{\ast }(x)=\frac{Np(x,x)}{N-sp(x,x)}$. Here, the nonlinearity f is in $C^1(\overline{\mathrm{\Omega }}\times \mathbb{R})$ and assumed to satisfy suitable assumptions. Using variational methods combined with monotonicity arguments, we obtain the existence of solutions to the problem in a fractional Sobolev space with variable exponent. To our best knowledge, this paper is the first attempt in the study of singular problems involving fractional $p(x,.)$-Laplace operators.

Keywords:

• Fractional $p(x,.)$-Laplace operators
• existence of solution
• singular equations
• variational methods
• generalized fractional Sobolev space

2010 Mathematics Subject Classifications:

• Primary 35J60
• secondary 35J91
• 35S30
• 46E35
• 58E30

Disclosure statement

No potential conflict of interest was reported by the author(s).