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

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