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Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems

  • * Corresponding author: Rushun Tian and Zhi-Qiang Wang

    * Corresponding author: Rushun Tian and Zhi-Qiang Wang 
This paper is supported by Beijing Natural Science Foundation (1174013), National Natural Science Foundation of China (11601353, 11771302, 11771324, 11671026, 11831009).
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  • In this paper, we study the following doubly coupled multicomponent system

    $ \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + \lambda_ju_j+ \sum_{k\neq j}\gamma_{jk}u_k = \mu_ju_j^3+ u_j\sum_{k\neq j}\beta_{jk}u_k^2,\\ u_j(x)\geq0\ \ \hbox{and}\ \ u_j\in H_0^1(\Omega), \end{array} \right. \end{equation*} $

    where $ \Omega\subset \mathbb{R} ^N $ and $ N = 2,3 $; $ \lambda_j, \gamma_{jk} = \gamma_{kj}, \mu_j, \beta_{jk} = \beta_{kj} $ are constants, $ j, k = 1, 2, ..., n $, $ n\geq 2 $. We prove some existence and nonexistence results for positive solutions of this system. If the system is fully symmetric, i.e. $ \lambda_j\equiv\lambda, \gamma_{jk}\equiv\gamma, \mu_j\equiv\mu, \beta_{jk}\equiv\beta $, we study the multiplicity and bifurcation phenomena of positive solution.

    Mathematics Subject Classification: 35B05, 35J61, 58C40, 35J15, 58E07.


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