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, cdot )-Laplacian problems involving potentials with vanishing behavior at infinity as an application.
Highlights
Let (M, g) be a smooth complete compact Riemannian n-manifold
The present paper is devoted to proving some qualitative properties of a new fractional Sobolev space with variable exponent in complete manifolds, as well as to studying the existence of weak solutions to the following problem as an application:
We prove a continuous and compact embedding theorem of this space into variable exponent Lebesgue spaces
Summary
The present paper is devoted to proving some qualitative properties of a new fractional Sobolev space with variable exponent in complete manifolds, as well as to studying the existence of weak solutions to the following problem as an application:. The motivation of this paper was, on the one hand, the work of Fu and Guo [24] who introduced the variable exponent function spaces on Riemannian manifolds in 2012, followed by Gaczkowski and Górka [25] who in 2013 examined the above space in the case of compact manifolds, and Guo [28] who in 2015 discussed the properties of the Nemytsky operator and obtained the existence of weak solutions for Dirichlet problems of nonhomogeneous p(m)-harmonic equations. For s ∈ (0, 1), we introduce the variable exponent Sobolev fractional space on a complete manifold as follows:.
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