Existence of positive solutions to elliptic problems involving the fractional Laplacian

摘要

This paper focuses on the following elliptic problem involving a fractional Laplacian: ( − Δ ) α u + V ( x ) u = f ( u ) in  R N , $$(-Delta)^{lpha}u+V(x)u=f(u) quad ext{in } mathbb{R}^{N}, $$ where N ≥ 2 $Ngeq2$ , α ∈ ( 0 , 1 ) $lphain(0,1)$ , ( − Δ ) α $(-Delta)^{lpha}$ stands for the fractional Laplacian. Using some variational methods, we obtain the existence of positive solutions without compactness conditions.