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arXiv · 1304.5030

Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations

Zhijie Chen·Chang-Shou Lin·Wenming Zou
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Abstract

We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-Δu_1 +\la_1 u_1 = μ_1 u_1^3+βu_1 u_2^2, \quad x\in Ω,\\ -Δu_2 +\la_2 u_2 =μ_2 u_2^3+βu_1^2 u_2, \quad x\in \Om,\\ u_1=u_2=0 \,\,\,\hbox{on \,$\partial\Om$}.\end{cases}\end{displaymath} Here $\Om\subset\RN (N=2, 3)$ is a smooth bounded domain, $\la_1, \la_2$, $μ_1, μ_2$ are all positive constants. We show that, for each $k\in\mathbb{N}$ there exists $\bb_k>0$ such that this system has at least $k$ sign-changing solutions (i.e., both two components change sign) and $k$ semi-nodal solutions (i.e., one component changes sign and the other one is positive) for each fixed $\bb\in (0, \bb_k)$.

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Zhijie Chen, Chang-Shou Lin, Wenming Zou. 2013-04-18. Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations. https://arxiv.org/abs/1304.5030

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