Compact embeddings for weighted fractional Sobolev spaces and applications to Nonlinear Schrödinger Equations

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 Schrödinger equation $$(-\mathrm{\Delta }{)}^{s}u+V(\mathrm{\mid }x\mathrm{\mid })u=K(\mathrm{\mid }x\mathrm{\mid })f(u),\text{ in }{\mathbb{R}}^N,$$ where the two measurable functions $K>0$ and $V\ge 0$ could vanish at infinity.