Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrodinger system

We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics: {displaymath} {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 \,$\partial\Om$}.{cases}{displaymath} Here $\Om$ is a smooth bounded domain in $\R^N (N=2, 3)$ or $\Om=\RN$, $\la_1,\, \la_2$, $μ_1,\,μ_2$ are all positive constants and the coupling constant $\bb<0$. We show that this system has infinitely many sign-changing solutions. We also obtain infinitely many semi-nodal solutions in the following sense: one component changes sign and the other one is positive. The crucial idea of our proof, which has never been used for this system before, is turning to study a new problem with two constraints. Finally, when $\Om$ is a bounded domain, we show that this system has a least energy sign-changing solution, both two components of which have exactly two nodal domains, and we also study the asymptotic behavior of solutions as $β\to -\infty$ and phase separation is expected.

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