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

Zhijie Chen, Chang-Shou Lin, Wenming Zou

• We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics:

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  −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.