Elliptic problems involving the fractional Laplacian in R^N

We study the existence and multiplicity of solutions for elliptic equations in R^N, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by (P), depends on a real parameter and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of \lambda which divide the real line in different intervals, where (P) admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions.