Elliptic problems involving the fractional Laplacian in RNRN

We study the existence and multiplicity of solutions for elliptic equations in RNRN, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by (Pλ)(Pλ), depends on a real parameter λ 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 λ   which divide the real line in different intervals, where (Pλ)(Pλ) admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions.