Standing Waves of Fractional Schrödinger Equations with Potentials and General Nonlinearities

We study the existence of standing waves of fractional Schrödinger equations with a potential term and a general nonlinear term: iuₜ − (−∆) ˢu − V(x)u + f(u) = 0, (t, x) ∈ R₊ × RN, where $s ∈ (0, 1),$ $N > 2s$ is an integer and $V(x) ≤ 0$ is radial. More precisely, we investigate the minimizing problem with L²-constraint: E(α)= inf\1/2∫RN|(-Δ)s/2u|²+V(x)|u|²-2F(|u|)\ | \ u∈ Hˢ(RN),||u||²L²(RN)=α\. Under general assumptions on the nonlinearity term $f(u)$ and the potential term $V(x),$ we prove that there exists a constant α₀ ≥ 0 such that $E(α)$ can be achieved for all α > α₀, and there is no global minimizer with respect to $E(α)$ for all 0 < α < α₀. Moreover, we propose some criteria determining α₀ = 0 or α₀ > 0.