Compact embeddings for weighted fractional Sobolev spaces and applications to Nonlinear Schr\"odinger Equations

TL;DR

This paper establishes a compact embedding for weighted fractional Sobolev spaces and applies it to prove the existence of solutions for a class of nonlinear Schrödinger equations using variational methods.

Contribution

It introduces a new compact embedding result for weighted fractional Sobolev spaces and applies it to nonlinear Schrödinger equations with variable potentials.

Findings

Proved a compact embedding theorem for weighted fractional Sobolev spaces.

Established existence of solutions for nonlinear Schrödinger equations with variable coefficients.

Applied variational methods to demonstrate solution existence under broad conditions.

Abstract

The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schr\"odinger equation $$ (-\Delta)^su + V(|x|)u = K(|x|)f(u), \quad \text{ in } \mathbb{R}^N, $$ where the two measurable functions $K > 0$ and $V \geq 0$ could vanish at infinity.