Structure of positive solutions to a class of Schrödinger systems

Xiyou Cheng; Zhitao Zhang

doi:10.3934/dcdss.2020461

Article Contents

2021, Volume 14, Issue 6: 1857-1870. Doi: 10.3934/dcdss.2020461

This issue Previous Article Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero Next Article A direct method of moving planes for fully nonlinear nonlocal operators and applications

Structure of positive solutions to a class of Schrödinger systems

Xiyou Cheng^1, and
Zhitao Zhang^2,

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
2.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received: May 31, 2020

Revised: August 31, 2020

Early access: November 2020

Published: June 2021

Supported in part by the NSFC (11771428, 11926335)

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Abstract

This paper is devoted to dealing with the existence and uniqueness of positive solutions for the following coupled nonlinear Schrödinger systems with multi-parameters

$ \begin{equation*} \begin{cases} &- \varDelta u = \lambda u - \mu_1 u^3 + \beta_1 uv^2,\quad \rm {in}\ \Omega,\\ &- \varDelta v = \lambda v - \mu_2 v^3 + \beta_2 u^2v,\quad \rm {in}\ \Omega,\\ &u, v > 0\quad \rm {in}\ \Omega,\quad u, v = 0\quad \rm {on}\ \partial \Omega, \end{cases} \end{equation*} $

on the range of $ \lambda $ and the different coupling constants $ \beta_1, \beta_2 $, where $ \Omega \subset \mathbb{R}^N $ $ (N \geqslant 1) $ is a bounded smooth domain, $ \lambda > 0 $ and $ \mu_1 \leqslant \mu_2 $. Under some conditions, we establish some interesting positive solutions structure theorems in the $ \beta_1 \beta_2 $-plane, especially we obtain the new structure theorems for the cases that $ \mu_1 $ and $ \mu_2 $ have different signs or they are negative. In addition, we get interesting uniqueness results via synchronized solutions techniques.

Keywords:

Mathematics Subject Classification: 35J10, 35J57, 35J25.

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Figure 1. structure (Ⅰ) of solutions for (1), with $\lambda > \lambda_0$

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Figure 2. structure (Ⅰ) of solutions for (1), with $0 < \lambda < \lambda_0$

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Figure 3. structure (Ⅰ) of solutions for (1), with $\lambda = \lambda_0$

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Figure 4. structure (Ⅱ) of solutions for (1), with $\lambda > \lambda_0$

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Figure 5. structure (Ⅱ) of solutions for (1), with $0 < \lambda < \lambda_0$

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Figure 6. structure (Ⅱ) of solutions for (1), with $\lambda = \lambda_0$

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