Uniqueness and nondegeneracy of ground states for the Schrödinger-Newton equation with power nonlinearity

Huxiao Luo

doi:10.3934/dcds.2024065

Article Contents

2024, Volume 44, Issue 11: 3443-3473. Doi: 10.3934/dcds.2024065

This issue Previous Article Exponential stability of solutions to the Schrödinger–Poisson equation Next Article Homogenization of attractors to reaction-diffusion system in a medium with random obstacles

Uniqueness and nondegeneracy of ground states for the Schrödinger-Newton equation with power nonlinearity

Huxiao Luo^,

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

^*Corresponding author: Huxiao Luo
^*Corresponding author: Huxiao Luo

Received: November 2023

Early access: May 2024

Published: November 2024

The first author is supported by [NSFC(11901532)].

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Abstract

In this article, we studied the Schrödinger-Newton equation combined with local nonlinearity

$ \begin{align} -\Delta u+\lambda u = \frac{1}{4\pi}\left(\frac{1}{|x|}\star u^{2}\right)u+|u|^{q-2}u \quad \text{in}\; \mathbb{R}^3, \end{align} ~~~~(1)$

where $ \lambda\in \mathbb{R}_+ $, $ q\in (2, 3)\cup(3, 6) $. First, we proved that the positive ground state of (1) tends toward the positive solution of the Schrödinger equation

$ \begin{align} -\Delta u+u = u^{q-1} \quad \text{in}\; \mathbb{R}^3, \end{align} ~~~~(2)$

or the Schrödinger-Newton equation

$ \begin{align} -\Delta u+u = \frac{1}{4\pi}\left(\frac{1}{|x|}\star u^{2}\right)u \quad \text{in}\; \mathbb{R}^3, \end{align}~~~~(3) $

as $ \lambda\to0^+ $ or $ \lambda\to+\infty $. By means of the uniqueness and nondegeneracy of positive solutions for the Schrödinger equation (2) and the Schrödinger-Newton equation (3), we proved the uniqueness of positive ground states for (1) when $ \lambda $ is sufficiently close to $ 0 $ or $ +\infty $. Moreover, by the action of the linearized equation with respect to decomposition into spherical harmonics, we obtained the nondegeneracy of ground states.

Keywords:

Schrödinger-Newton equation,
uniqueness and nondegeneracy of ground states,
spherical harmonics.

Mathematics Subject Classification: Primary: 35A02, 35B20; Secondary: 35J15, 35J20, 35J61.

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