On a class of nonlocal problems in new fractional Musielak-Sobolev spaces

ABSTRACT

In this paper, we first introduce the new fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces. Then, using the direct variational approach, we investigate the existence of a nontrivial weak solution for a class of fractional type problems with Dirichlet boundary data of the following form $\left({\mathcal{P}}_a\right)\left\{\begin{array}{cc}(-\mathrm{\Delta }{)}_{a_{(x,.)}}^{s}u+{\hat{a}}_x(|u|)u=f(x,u)& \text{in }\mathrm{\Omega },\\ u=0& \text{in }{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}\right.$ where $s\in (0,1)$, Ω is an open bounded subset in ${\mathbb{R}}^N$ with Lipschitz boundary $\mathrm{\partial }\mathrm{\Omega }$, and $f:\mathrm{\Omega }\times \mathbb{R}⟶\mathbb{R}$ is a Carathéodory function with suitable growth conditions.

KEYWORDS:

• Fractional Musielak-Sobolev spaces
• Nonlocal problems
• Dirichlet problem
• Direct variational method

2010:

• 46E35
• 35R11
• 35J20
• 47G20

Acknowledgements

The authors gratefully acknowledge the reviewer(s) for their valuable comments and suggestions which have improved the quality of this paper.

Disclosure statement

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