Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential

Leijin Cao

doi:10.3934/dcdsb.2022125

Article Contents

2023, Volume 28, Issue 2: 1342-1366. Doi: 10.3934/dcdsb.2022125

This issue Previous Article Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem Next Article An adaptive finite element method for the elastic transmission eigenvalue problem

Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential

Leijin Cao^,

School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

^* Corresponding author: Leijin Cao
^* Corresponding author: Leijin Cao

Received: August 2021

Revised: May 2022

Early access: July 2022

Published: February 2023

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Abstract

In this paper, we consider the existence of stable standing waves for the nonlinear Schrödinger equation with combined power nonlinearities and the Hardy potential. In the $ L^2 $-critical case, we show that the set of energy minimizers is orbitally stable by using concentration compactness principle. In the $ L^2 $-supercritical case, we show that all energy minimizers correspond to local minima of the associated energy functional and we prove that the set of energy minimizers is orbitally stable.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J20.

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