Standing Waves of Fractional Schrödinger Equations with Potentials and General Nonlinearities

Abstract

We study the existence of standing waves of fractional Schrödinger equations with a potential term and a general nonlinear term: $$iu_t − (−∆) ^su − V(x)u + f(u) = 0, (t, x) ∈ \mathbb{R}_+ × \mathbb{R}^N,$$ where $s ∈ (0, 1),$ $N > 2s$ is an integer and $V(x) ≤ 0$ is radial. More precisely, we investigate the minimizing problem with $L^2$-constraint: $$E(\alpha)={\rm inf}\left\{\frac{1}{2}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2+V(x)|u|^2-2F(|u|)\mid u\in H^s(\mathbb{R}^N),||u||^2_{L^2(\mathbb{R}^N)}=\alpha\right\}.$$ Under general assumptions on the nonlinearity term $f(u)$ and the potential term $V(x),$ we prove that there exists a constant $α_0 ≥ 0$ such that $E(α)$ can be achieved for all $α > α_0,$ and there is no global minimizer with respect to $E(α)$ for all $0 < α < α_0.$ Moreover, we propose some criteria determining $α_0 = 0$ or $α_0 > 0.$