二维Schrodinger-Newton方程最小能量奇解的存在性与轴对称性

Existence and Axial Symmetry of Minimal Action Odd Solutions for 2-D Schrodinger-Newton Equation

在本文中,对所有的p≥2,我们考虑如下的二维空间R^(2)上的Schrodinger-Newton方程{-△u+u=w|u|^(p-1) -△w=2π|u|^(p),使用变分方法和Cerami紧性性质,我们证明存在最小能量的奇对称解.同时,对上半空间上的一个相似但是更加复杂的方程,使用移动平面法,证明这些奇对称解事实上是轴对称的.我们的结果,可以部分地看作文献[13]在二维空间上的相应的结果,也可以看作是文献[10]推广到奇解的情形.

We consider the following 2-D Schrodinger-Newton equation {-△u+u=w|u|^(p-1) -△w=2π|u|^(p),in R^(2) for all p≥2.Using variational method with the Cerami compactness property,we prove the existence of minimal action odd solutions.Also by carefully applying the method of moving plane to a similar but more complex equation on the upper half space,we prove these solutions are in fact axially symmetric.Our results can be seen partially as the counterpart of the results in[13]for the 2-D case,or the extension of the results in[10]to the odd solution case.