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 results. Moreover, we prove that any function in $W^s{L}_{{\Phi }_{x,y}}\left(\mathrm{\Omega }\right)$ may be extended to a function in $W^s{L}_{{\Phi }_{x,y}}\left({\mathbb{R}}^N\right)$, with $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain of class $C^{0,1}$. In addition, we establish a result that 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.

Keywords:

• Fractional Musielak–Sobolev spaces
• Nonlocal problems
• Direct variational method

2010 Mathematics Subject Classifications:

• 46E35
• 35R11
• 35J20
• 47G20

Disclosure statement

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