Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials

Yali Meng; Yali Meng

doi:10.3934/math.2022332

AIMS Mathematics

2022, Volume 7, Issue 4: 5957-5970. doi: 10.3934/math.2022332

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Research article

Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials

Yali Meng ^,

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received: 03 December 2021 Revised: 05 January 2022 Accepted: 09 January 2022 Published: 14 January 2022
MSC : 35Q55

In this paper, we consider the following nonlinear Schrödinger equation with attractive inverse-power potentials

$ i\partial_t\psi+\Delta\psi+\gamma|x|^{-\sigma}\psi+|\psi|^\alpha\psi = 0, \; \; \; (t, x)\in\mathbb{R}\times\mathbb{R}^N, $

where $ N\geq3 $, $ 0 < \gamma < \infty $, $ 0 < \sigma < 2 $ and $ \frac{4}{N} < \alpha < \frac{4}{N-2} $. By using the concentration compactness principle and considering a local minimization problem, we prove that there exists a $ \gamma_0 > 0 $ sufficiently small such that $ 0 < \gamma < \gamma_0 $ and for any $ a\in(0, a_0) $, there exist stable standing waves for the problem in the $ L^2 $-supercritical case. Our results are complement to the result of Li-Zhao in ^[23].
- nonlinear Schrödinger equation,
- inverse-power potentials,
- stable standing waves
Citation: Yali Meng. Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials[J]. AIMS Mathematics, 2022, 7(4): 5957-5970. doi: 10.3934/math.2022332

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Abstract

In this paper, we consider the following nonlinear Schrödinger equation with attractive inverse-power potentials

$ i\partial_t\psi+\Delta\psi+\gamma|x|^{-\sigma}\psi+|\psi|^\alpha\psi = 0, \; \; \; (t, x)\in\mathbb{R}\times\mathbb{R}^N, $

where $ N\geq3 $, $ 0 < \gamma < \infty $, $ 0 < \sigma < 2 $ and $ \frac{4}{N} < \alpha < \frac{4}{N-2} $. By using the concentration compactness principle and considering a local minimization problem, we prove that there exists a $ \gamma_0 > 0 $ sufficiently small such that $ 0 < \gamma < \gamma_0 $ and for any $ a\in(0, a_0) $, there exist stable standing waves for the problem in the $ L^2 $-supercritical case. Our results are complement to the result of Li-Zhao in ^[23].

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