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

We study the following coupled Schrodinger equations which have appeared as several models from mathematical physics: { -Delta u(1) + lambda(1)u(1) = mu(1)u(1)(3) +beta u(1)u(2)(2) x epsilon Omega -Delta u(2) + lambda(2)u(2) = mu(2)u(2)(3) +beta u(1)(2)u(2) x epsilon Omega u(1) = u(2) = 0 on partial derivative Omega Here Omega is a smooth bounded domain in R-N (N = 2, 3) or Omega =R-N ,lambda(1), lambda(2,) mu(1), mu(2) are all positive constants and the coupling constant beta < 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 to study a new problem with two constraints. Finally, when Omega 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 -> -infinity and phase separation is expected.