Sign-changing solutions for Schrödinger system with critical growth

Changmu Chu; Jiaquan Liu; Zhi-Qiang Wang; Changmu Chu; Jiaquan Liu; Zhi-Qiang Wang

doi:10.3934/era.2022013

Electronic Research Archive

2022, Volume 30, Issue 1: 242-256. doi: 10.3934/era.2022013

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Sign-changing solutions for Schrödinger system with critical growth

1.
School of Preparatory Education, Guizhou Minzu University, Guizhou 550025, China
2.
LMAM, School of Mathematical Science, Peking University, Beijing 100871, China
3.
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

Received: 13 August 2021 Revised: 22 October 2021 Accepted: 22 October 2021 Published: 28 December 2021

We consider the following Schrödinger system

$ \begin{equation*} \left\{ \begin{aligned} &-\Delta u_j = \sum\limits_{i = 1}^k \beta_{ij}|u_i|^3|u_j|u_j+\lambda_j|u_j|^{q-2}u_j, \ \ \ \text{in}\, \, \Omega, \\ &u_j = 0\quad\text{on}\, \, \partial\Omega, \, \, j = 1, \cdots, k \end{aligned} \right. \end{equation*} $

where $ \Omega\subset\mathbb{R}^3 $ is a bounded domain with smooth boundary. Assume $ 5 < q < 6, \, \lambda_j > 0, \, \beta_{jj} > 0, \, j = 1, \cdots, k $, $ \beta_{ij} = \beta_{ji}, \, i\neq j, i, j = 1, \cdots, k $. Note that the nonlinear coupling terms are of critical Sobolev growth in dimension $ 3 $. We prove that under an additional condition on the coupling matrix the problem has infinitely many sign-changing solutions. The result is obtained by combining the method of invariant sets of descending flow with the approach of using approximation of systems of subcritical growth.
- Schrödinger system,
- critical growth,
- sign-changing solutions
Citation: Changmu Chu, Jiaquan Liu, Zhi-Qiang Wang. Sign-changing solutions for Schrödinger system with critical growth[J]. Electronic Research Archive, 2022, 30(1): 242-256. doi: 10.3934/era.2022013

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Abstract

We consider the following Schrödinger system

$ \begin{equation*} \left\{ \begin{aligned} &-\Delta u_j = \sum\limits_{i = 1}^k \beta_{ij}|u_i|^3|u_j|u_j+\lambda_j|u_j|^{q-2}u_j, \ \ \ \text{in}\, \, \Omega, \\ &u_j = 0\quad\text{on}\, \, \partial\Omega, \, \, j = 1, \cdots, k \end{aligned} \right. \end{equation*} $

where $ \Omega\subset\mathbb{R}^3 $ is a bounded domain with smooth boundary. Assume $ 5 < q < 6, \, \lambda_j > 0, \, \beta_{jj} > 0, \, j = 1, \cdots, k $, $ \beta_{ij} = \beta_{ji}, \, i\neq j, i, j = 1, \cdots, k $. Note that the nonlinear coupling terms are of critical Sobolev growth in dimension $ 3 $. We prove that under an additional condition on the coupling matrix the problem has infinitely many sign-changing solutions. The result is obtained by combining the method of invariant sets of descending flow with the approach of using approximation of systems of subcritical growth.

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