Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations
arXiv:1304.5030
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, $\la_1, \la_2$, are all positive constants. We show that, for each there exists $\bb_k>0$ such that this system has at least sign-changing solutions (i.e., both two components change sign) and semi-nodal solutions (i.e., one component changes sign and the other one is positive) for each fixed $\bb\in (0, \bb_k)$.
This work continues the study of arXiv:1212.3773. Any comment is welcome