Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrodinger system
arXiv:1212.3773
Abstract
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 or $\Om=\RN$, $\la_1,\, \la_2$, 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 and phase separation is expected.
Final version, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sic