Standing Waves of Fractional Schrödinger Equations with Potentials and General Nonlinearities
Year: 2023
Author: Zaizheng Li, Qidi Zhang, Zhitao Zhang
Analysis in Theory and Applications, Vol. 39 (2023), Iss. 4 : pp. 357–377
Abstract
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{12∫RN|(−Δ)s2u|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 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.OA-2022-0012
Analysis in Theory and Applications, Vol. 39 (2023), Iss. 4 : pp. 357–377
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Fractional Schrödinger equation standing wave normalized solution.
Author Details
Zaizheng Li Email
Qidi Zhang Email
Zhitao Zhang Email