On the fractional Musielak-Sobolev spaces in [formula omitted]: Embedding results & applications

Abstract

This paper deals with new continuous and compact embedding theorems for the fractional Musielak-Sobolev spaces in Rd. As an application, using the variational methods, we obtain the existence of a nontrivial weak solution for the following Schrödinger equation(−Δ)gx,ysu+V(x)g(x,x,u)=b(x)|u|p(x)−2u,for allx∈Rd, where (−Δ)gx,ys is the fractional Museilak gx,y-Laplacian, V is a potential function, b∈Lδ′(x)(Rd), and p,δ∈C(Rd,(1,+∞))∩L∞(Rd). We would like to mention that the theory of the fractional Musielak-Sobolev spaces is in a developing state and there are few papers on this topic, see [6,11,12]. Note that, all these latter works dealt with the bounded case and there are no results devoted for the fractional Musielak-Sobolev spaces in Rd. Since the embedding results are crucial in applying variational methods, this work will provide a bridge between the fractional Mueislak-Sobolev theory and PDE's.