Existence of positive solutions to elliptic problems involving the fractional Laplacian

This paper focuses on the following elliptic problem involving a fractional Laplacian: (−Δ)αu+V(x)u=f(u)in RN,$$(-\Delta)^{\alpha}u+V(x)u=f(u) \quad \text{in } \mathbb{R}^{N}, $$ where N≥2$N\geq2$, α∈(0,1)$\alpha\in(0,1)$, (−Δ)α$(-\Delta)^{\alpha}$ stands for the fractional Laplacian. Using some variational methods, we obtain the existence of positive solutions without compactness conditions.