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

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