含扰动项的Chern-Simons-Schrödinger方程的多解  

Multiple solutions to the Chern-Simons-Schrödinger equation with perturbation

作　　者：元浩 翁立夫 周焕松 Hao Yuan;Lifu Weng;Huansong Zhou

机构地区：[1]武汉理工大学数学科学研究中心,武汉430070

出　　处：《中国科学：数学》2023年第9期1213-1226,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11931012和11871387)资助项目。

摘　　要：本文研究二维空间R2上一类含有Chern-Simons型非局部项且具有p−1(2<p<+∞)次幂次增长非线性项的Chern-Simons-Schrodinger方程在扰动项g(x)作用下多解的存在性.当g(x)≡0时,已有文献的结果表明,关于这类方程解的存在性研究,会因为所采用方法的不同而对幂次p的取值范围有不同的要求,并对相关参数还有一定的限制.当g(x)■0且其L^(2)范数小于某个可显式给出的确定值时,本文对于任意的p∈(2,∞)建立这类方程解的存在性或非存在性,并且对于适当范围内的p还得到这类方程同时具有正能量解和负能量解.We study the existence of solutions to the Chern-Simons-Schrodinger equation with a power type nonlinearity of exponent p−1(p>2)and a perturbation term g(x).When g(x)≡0,it is known that the methods for getting solutions to this kind of equation depend heavily on the values of the exponent p and certain restrictions for related parameters.In this paper,we study the case of g(x)≡0,and we establish the existence of solutions to the equation for any p>2 when the L^(2) norm of g(x)is less than a given constant which can be given precisely.In particular,for some range of p,we also obtain two solutions,with positive energy and negative energy,respectively,to the equation.

关 键 词：椭圆型偏微分方程 临界点 山路引理 Chern-Simons-Schrödinger方程 Ekeland变分引理 

分 类 号：O175[理学—数学]