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: {-δu1 + λ1u1 = μ1u31 + βu1u22 x ϵ ω-δu2 + λ2u2 = μ2u32 + βu1u22 x ϵ ωU1=U2 on δω} on Here ω is a smooth bounded domain in RN(N = 2, 3) or ω2 = RN, λ1, λ2, μ1,μ2, all positive constants and the coupling constant β < 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 £2 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 → - ∞ and phase separation is expected.