Embedding and extension results in Fractional Musielak-Sobolev spaces

Abstract

In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces $W^s{L}_{{\Phi }_{x,y}}$ such that the generalized Poincaré type inequality and some continuous and compact embedding theorems of these spaces. Moreover, we prove that any function in $W^s{L}_{{\Phi }_{x,y}}(\Omega )$ may be extended to a function in $W^s{L}_{{\Phi }_{x,y}}({\mathbb{R}}^N)$, with $\Omega \subset {\mathbb{R}}^N$ is a bounded domain of class $C^{0,1}$. In addition, we establish a result relates to the complemented subspace in $W^s{L}_{{\Phi }_{x,y}}\left({\mathbb{R}}^N\right)$. As an application, using the mountain pass theorem and some variational methods, we investigate the existence of a nontrivial weak solution for a class of nonlocal fractional type problems with Dirichlet boundary data.