On a new fractional Sobolev space with variable exponent on complete manifolds

Abstract

Abstract We present the theory of a new fractional Sobolev space in complete manifolds with variable exponent. As a result, we investigate some of our new space’s qualitative properties, such as completeness, reflexivity, separability, and density. We also show that continuous and compact embedding results are valid. We apply the conclusions of this study to the variational analysis of a class of fractional p ( z , ⋅ ) $p(z, \cdot )$ -Laplacian problems involving potentials with vanishing behavior at infinity as an application.

Keywords

• Fractional p ( z , ⋅ ) $p( z, \cdot )$ -Laplacian
• Existence of solutions
• Fractional Sobolev space with variable exponent on complete manifolds
• Variational method