(untitled) (Analysis in Theory and Applications, 2023)
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) ∈ R_+ × 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(α)=inf{12_R^N|(-Δ)^s2 u|^2+V(x)|u|^2-2F(|u|) | uH^s(R^N),||u||^2_L^2(R^N)=α}. 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.