Chern-Simons-SchrOdinger方程能量泛函的L^(2)约束极小化问题  

Research on the Lowest Energy Solution of Chern-Simons-Schr?dinger Equation with Trapping Potential

作　　者：杨迎 沈烈军 Yang Ying;Shen Liejun(Center of Mathematics,Wuhan University of Technology,Wuhan 430070)

机构地区：[1]武汉理工大学数学科学研究中心,武汉430070

出　　处：《数学物理学报（A辑）》2022年第3期716-729,共14页Acta Mathematica Scientia

基　　金：国家自然科学基金(11931012,11871387)~。

摘　　要：该文主要研究R^(2)上一类Chern-Simons-SchrOdinger(CSS)方程在给定L^(2)范数下解的存在性.这类问题可转化为该方程对应能量泛函E_(p)^(β)(u)在约束条件‖u‖L^(2)(R^(2))=1下的变分求极小问题.对于质量次临界的情形,即p∈(0,2),该文应用简洁的方法证明了无论位势函数V(x)是否为0,这类约束变分极小化问题都是可达的;对于质量临界的情形,即p=2,该文找到了两个可显式给出的正常数β^(*)>β_(*),使得V((x)≡0时的约束变分极小化问题对于β>β_(*)或β∈(0,β_(*)]均不可达,而对于V(x)≠0时的约束变分极小化问题则在β∈(0,β_(*)]可达,β>β_(*)不可达.此外,该文还讨论了质量次临界的约束极小能量在p→2时的极限行为.In this paper,we mainly study the existence of solutions with prescribed L^(2)-norm to the Chern-Simons-Schrodinger(CSS)equation.This type problem can be transformed into look for the minimizer of the corresponding energy functional E;(u)under the constraint‖u‖L^(2)(R^(2))=1.Concerning the subcritical mass case,that is,p∈(0,2),no matter whether the potential function V(x)equals to 0,we prove that the constraint minimization can be achieved by some simple methods.We are also concerned with the critical mass case of p=2:if V(x)≡0,there exist two constantsβ^(*)>β_(*)>0 which can be explicitly determined such that the constraint minimization cannot achieved for anyβ∈(0,β_(*)]∪(β*,+∞);if V(x)■0,the constraint minimization cannot be achieved forβ>β_(*),but can be achieved forβ∈(0,β_(*)].In addition,we discuss the limit behavior of the mass subcritical constrained minimum energy when p■2.

关 键 词：Chern-Simons-SchrOdinger方程 能量估计 约束变分 极限行为 

分 类 号：O175.2[理学—数学]