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

We study the following coupled Schr"{o}dinger equations which have appeared as several models from mathematical physics: {displaymath} {cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega, -\Delta u_2 +\la_2 u_2 =\mu_2 u_2^3+\beta u_1^2 u_2, \quad x\in \Om, u_1=u_2=0 ,,,\hbox{on ,$\mathrm{\partial }\text{Om}$}.{cases}{displaymath} Here $\text{Om}$ is a smooth bounded domain in ${\mathbb{R}}^N(N=2,3)$ or $\text{Om}=\text{RN}$, ${\text{la}}_1,{\text{la}}_2$, ${\mu }_1,{\mu }_2$ are all positive constants and the coupling constant $\text{bb}<0$. 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 $\text{Om}$ 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 $\beta \to -\mathrm{\infty }$ and phase separation is expected.