首页|Standing Waves of Fractional Schr?dinger Equations with Potentials and General Nonlinearities

Standing Waves of Fractional Schr?dinger Equations with Potentials and General Nonlinearities

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We study the existence of standing waves of fractional Schrödinger equa-tions 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|(-△)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 α0 ≥ 0 such that E(α)can be achieved for allα＞αo,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.

Fractional Schrödinger equationstanding wavenormalized solution

Zaizheng Li、Qidi Zhang、Zhitao Zhang

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School of Mathematical Sciences,Hebei Normal University,Shijiazhuang,Hebei 050024,China

Hebei Center for Applied Mathematics,Shijiazhuang,Hebei 050024,China

Department of Mathematics,The University of Hong Kong,Hong Kong,China

School of Mathematical Sciences,Jiangsu University,Zhenjiang,Jiangsu 212013,China

HLM,Academy of Mathematics and Systems Science,the Chinese Academy of Sciences,Beijing 100190,China

School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

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Natural Science Foundation of Hebei Provincescience and Technology Project of Hebei Education Departmentscience Foundation of Hebei Normal UniversityNational Natural Science Foundation of ChinaNational Natural Science Foundation of ChinaNational Natural Science Foundation of China

A2022205007QN2022047L2021B05117714281203101512026217

2023

10.4208/ata.OA-2022-0012

分析、理论与应用(英文版)
南京大学

分析、理论与应用(英文版)

影响因子：0.111
ISSN：1672-4070
年,卷(期)：2023.39(4)
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