Existence and axial symmetry of minimal action odd solutions for 2-D Schr\"{o}dinger-Newton equation

TL;DR

This paper proves the existence of minimal action odd solutions for a 2-D Schr"{o}dinger-Newton equation and demonstrates their axial symmetry using variational methods and the moving plane technique.

Contribution

It establishes the existence and symmetry properties of odd solutions for the 2-D Schr"{o}dinger-Newton equation, extending previous results to the odd solution case.

Findings

Existence of minimal action odd solutions.

Solutions are axially symmetric.

Extension of previous 2-D Schr"{o}dinger-Newton results.

Abstract

We consider the following 2-D Schr\"{o}dinger-Newton equation \begin{eqnarray*} \begin{cases} -\Delta u+u=w|u|^{p-1}u \\ -\Delta w=2 \pi |u|^p \end{cases}\text{in} \; \mathbb{R}^2 \end{eqnarray*} for $ p \geq 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 partially can be seen as the counterpart of results in paper \cite{GS} for the 2-D case, or the extension of the results \cite{CW} to the odd solution case.