Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian

Abstract

The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation ${\Delta }^{\frac{\alpha }{2}}u=\rho (x)\phi (u)$ admits nonnegative nontrivial bounded solutions in the whole space ${\mathbb{R}}^N$.