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

Abstract

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 $$ (-\Delta)^s u + V(x)u=Q(x)|u|^{p-2}u \quad\text{in }\mathbb{R}^N,\ N\geq 1,$$ where the exponent $p$ is superlinear but subcritical, and $V>0$, $Q\geq 0$ are bounded functions converging to $1$ as $|x|\to\infty$. 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 $Q$.