Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction
Zhitao Zhang Haijun Luo
Communications on Pure & Applied Analysis 2018, 17(3): 787-806 doi: 10.3934/cpaa.2018040
We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrödinger elliptic system
$\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$
$ 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.
keywords: Nonlinear elliptic system ground state solution foliated Schwarz symmetric asymptotic limits

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