On the Symmetry of the Ground States of Nonlinear Schrödinger Equation with Potential
On the Symmetry of the Ground States of Nonlinear Schrödinger Equation with Potential
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DOI：
10.1515/ans-2010-0409
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发表时间：
2010-11
影响因子：
1.8
通讯作者：
Masaya Maeda
中科院分区：
文献类型：
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作者：
Masaya Maeda
摘要研究了L2范数约束下能量泛函的极小化问题。我们证明了当L2-范数较小时，极小元是唯一的，而当L2-范数较大时，极小元集中在b的最大值点处，并且指数衰减。根据这一结果，我们可以证明，如果V和b是径向对称的，但b在原点没有达到最大值，则对称破缺随着L2范数的增加而发生。进一步地，我们证明了当b有多个极大值点时，极小值点集中在由b，V定义的函数和-∆φ+φ-φp=0的唯一正径向解的极小点上。对于V和b是径向对称的情形，我们证明了如果极小元集中在原点，则极小元是径向对称的。进一步，我们构造了一个能量泛函，使得极小化子一次破坏它的对称性，但之后随着L2范数的增加，它又恢复到对称。
Abstract We investigate the minimizers of the energy functional under the constraint of the L2-norm. We show that for the case L2-norm is small, the minimizer is unique and for the case L2-norm is large, the minimizer concentrate at the maximum point of b and decays exponentially. By this result, we can show that if V and b are radially symmetric but b does not attain its maxi- mum at the origin, then the symmetry breaking occurs as the L2-norm increases. Further, we show that for the case b has several maximum points, the minimizer concentrates at a point which minimizes a function which is defined by b, V and the unique positive radial solution of -∆φ + φ - φp = 0. For the case when V and b are radially symmetric, we show that if the minimizer concentrates at the origin, then the minimizer is radially symmetric. Further, we construct an energy functional such that the minimizer breaks its symmetry once but after that it recovers to be symmetric as the L2-norm increases.