Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

Yile Wang; Yile Wang

doi:10.3934/math.2021345

AIMS Mathematics

2021, Volume 6, Issue 6: 5837-5850. doi: 10.3934/math.2021345

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Research article

Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

Yile Wang ^,

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

Received: 25 January 2021 Accepted: 17 March 2021 Published: 26 March 2021
MSC : 35Q55

In this paper, we investigate the existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

$ i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi = 0,\; \; (t,x)\in [0,T^\star)\times \mathbb{R}^N. $

By using concentration compactness principle, when one nonlinearity is focusing and $ L^2 $-critical, the other is defocusing and $ L^2 $-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper ^[14] to the $ L^2 $-critical and $ L^2 $-supercritical nonlinearities.
- concentration compactness principle,
- orbital stability,
- inverse-power potential,
- standing waves
Citation: Yile Wang. Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities[J]. AIMS Mathematics, 2021, 6(6): 5837-5850. doi: 10.3934/math.2021345

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Abstract

In this paper, we investigate the existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

$ i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi = 0,\; \; (t,x)\in [0,T^\star)\times \mathbb{R}^N. $

By using concentration compactness principle, when one nonlinearity is focusing and $ L^2 $-critical, the other is defocusing and $ L^2 $-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper ^[14] to the $ L^2 $-critical and $ L^2 $-supercritical nonlinearities.

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