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 (Pa)(−Δ)a(x,.)su+aˆx(|u|)u=f(x,u)in Ω,u=0in RN∖Ω, where s∈(0,1), Ω is an open bounded subset in RN with Lipschitz boundary ∂Ω, and f:Ω×R⟶R is a Carathéodory function with suitable growth conditions.

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