Positive solutions to some nonlinear fractional Schr\"odinger equations via a min-max procedure

The existence of a positive solution to the following fractional semilinear equation is proven, in a situation where a ground state solution may not exist. More precisely, we consider for $0<s<1$ the equation $$(-\mathrm{\Delta }{)}^{s}u+V(x)u=Q(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u\text{in }{\mathbb{R}}^N,N\ge 1,$$ where the exponent $p$ is superlinear but subcritical, and $V>0$, $Q\ge 0$ are bounded functions converging to $1$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. Using a min-max procedure introduced by Bahri and Li we prove the existence of a positive solution under one-sided asymptotic bounds for $V$ and $Q$.