Existence and axial symmetry of minimal action odd solutions for 2-D Schrödinger-Newton equation

2019/12/06 by Yang Zhang, Zhang, Yang
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1912.03060

openalex publication_date 2019/12/06 · openalex created_date 2019/12/13 · openalex updated_date 2026/07/28

Abstract

We consider the following 2-D Schrödinger-Newton equation \begincases -Δu+u=w|u|^p-1u
-Δw=2 π|u|^p \endcasesin ℝ² for 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 partially can be seen as the counterpart of results in paper \citeGS for the 2-D case, or the extension of the results \citeCW to the odd solution case.

Citations

• Bound states of two-dimensional Schrödinger-Newton equations
• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• Nodal solutions for the Choquard equation
• Uniqueness theorems through the method of moving spheres
• Classification of solutions for an integral equation
• On Gravity's role in Quantum State Reduction
• The Schrödinger–Poisson equation under the effect of a nonlinear local term

Related

• Existence of axially symmetric solutions in SU(2)-Yang-Mills and related theories
• Bound states of two-dimensional Schrödinger-Newton equations
• Existence and uniqueness of ground state solutions for the planar Schrödinger-Newton equation on the disc
• Bounds states of the Schrödinger-Newton model in low dimensions
• Complex-valued Solutions of the Planar Schrödinger-Newton System