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Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction

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Supported by the National Natural Science Foundation of China 11325107,11331010,11771428
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  • We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrödinger elliptic system

    $\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$

    where $ N≤3, Ω\subseteq\mathbb{R}^N$ is a smooth domain. First we establish the symmetry of ground state solutions, that is, when $ Ω$ is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that $ Ω$ is a ball or the whole space $ \mathbb{R}^N$. Next we investigate the asymptotic behavior of positive ground state solution as $ κ\to 0^-$, which shows that the limiting profile is exactly a minimizer for $ c_0$ (the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.

    Mathematics Subject Classification: Primary: 35J20, 35J61; Secondary: 35P30.


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