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

Abstract

We study the following coupled Schrodinger equations which have appeared as several models from mathematical physics: 8>>< >>: −1u1 + 1u1 = μ1u31 + #u1u22 x 2  −1u2 + 2u2 = μ2u32 + #u21 u2 x 2  u1 = u2 = 0 on @. Here  is a smooth bounded domain in RN (N = 2, 3) or  = RN , 1, 2, μ1, μ2 are all positive constants and the coupling constant # < 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  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 # ! −1 and phase separation is expected.