一类非线性薛定谔方程的涡环解

Vortex ring solutions of a class of nonlinear Schrodinger equations

本文考虑一类非线性薛定谔方程的涡环解.特别地,对二维非线性薛定谔方程,当径向磁场,电场满足恰当的符号条件时,证明了其存在任意指定的零点个数的非径向解,并且该解在无穷远处指数衰减.特别地,库伦位势,反平方位势,阶梯形位势等经典物理情形满足本文的条件.为了证明主要结果,除了使用靶向法处理约化的常微分方程外,本文主要引进了一个新的Pohozaev恒等式和辅助泛函,以及若干恰当函数变换.

The paper considers the vortex ring solutions of the nonlinear Schrödinger equation.For the nonlinear Schrödinger equation in two dimensions,we prove that the equations have a non-radial solution with arbitrarily specified number of zeros which decays exponentially at infinity,when the radial magnetic and electric potential satisfy appropriate sign conditions.In particular,classical physical situations such as the Coulomb potentials,inverse square potentials and steplike potentials satisfy our conditions.In order to prove the main result,besides using the shooting method to deal with the reduced ordinary differential equations,we mainly introduce a new Pohozaev identity,auxiliary functional and some appropriate function transformations.