On the generalized fractional Sobolev spaces and applications

Abstract

In this thesis, we study some generalizations of the fractional order Sobolev spaces and applications. Specifically, in the case of the fractional Orlicz-Sobolev spaces, we present an overview of recent developments in the theory, focusing on qualitative properties and embedding results. We then apply these results, along with the nonlinear Rayleigh quotient method and the minimization method on the Nehari manifold, to investigate conditions that ensure the existence of nontrivial solutions to a class of superlinear fractional Φ-Laplacian type problems with two parameters. In the context of fractional Musielak-Sobolev spaces, we extend and complement the existing theoretical results. More precisely, we establish some abstract results, such as uniform convexity, the Radon-Riesz property with respect to the modular function, the (S+)-property, a Brezis-Lieb type lemma for the modular function, and monotonicity results. Moreover, we apply the developed theory to study the existence of solutions to a class of problems involving a general nonlocal nonlinear operator of the fractional Φ-Laplacian type. Finally, we study the asymptotic behavior of modular functions and seminorms associated with fractional Musielak-Sobolev spaces as the fractional parameter approaches 1, without requiring the Δ2-condition on the Musielak function or its complementary function. This investigation culminates in a Bourgain-Brezis-Mironescu type formula for a very general family of functionals. It is important to emphasize that the achieving these results required the introduction of specific assumptions regarding the Musielak functions involved.