Description

• Title:
  Sign-changing solutions for coupled Schrödinger equations with mixed coupling
• Creator:
  Sato, Yohei;
  Wang, Zhi-Qiang
• Description:
  We consider the following nonlinear elliptic systems: \begin{equation*} \left\{\begin{alignedat}2 -\Delta u_1 +\lambda_1u_1 & = \mu_1u_1^3+\beta_{12}u_1u_2^2+\beta_{13}u_1u_3^2 &\quad& \hbox{in } \Omega,\\ -\Delta u_2 +\lambda_2u_2 & = \mu_2u_2^3+\beta_{12}u_2u_1^2+\beta_{23}u_2u_3^2 &\quad &\hbox{in } \Omega,\\ -\Delta u_3 +\lambda_3u_3 & = \mu_3u_3^3+\beta_{13}u_3u_1^2+\beta_{23}u_3u_2^2 &\quad& \hbox{in } \Omega,\\ \overrightarrow{u} & =(u_1,u_2,u_3)\in H^1_0(\Omega)^3, \end{alignedat}\right. \end{equation*} where $\Omega\subset \R^n$ ($n\leq 3$) is a bounded domain with smooth boundary, $\lambda_j, \mu_j> 0$ $(j=1,2,3)$, $\beta_{12}> 0$, and $\beta_{13}, \beta_{23}\leq 0$. For this model case of coupled Schrödinger equations with mixed coupling, for sufficiently large $\beta_{12}> 0$, we show that there exists a solution $(u_{1\beta},u_{2\beta},u_{3\beta})$ of $(*)$ such that $u_{1\beta}> 0, u_{2\beta}> 0$ and $u_{3\beta}$ changes sign exactly once. We also show that, for any given $k\in \mathbb{N}$, there exist $k$ vector solutions $(u_{1\beta}^\ell,u_{2\beta}^\ell,u_{3\beta}^\ell)$ $(\ell=1,\ldots,k)$ and these solutions are characterized by the genus with respect to a partial symmetry $\sigma(u_1,u_2,u_3)=(-u_1,-u_2,u_3)$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2020 Topological Methods in Nonlinear Analysis
• Date:
  2020.12.18
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 57, No 1 (March 2021); 243 - 274;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2020.016/27448 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2020.016/27449
• Subject:
  Nonlinear elliptic systems;
  mixed coupling;
  sign-changing solutions;
  Primary;
  35J57;
  35J50;
  Secondary;
  35J91
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2020.016
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other