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

• Zhijie Chen ^[1] ; Chang-Shou Lin ^[2] ; Wenming Zou ^[1]
  1. [1] Tsinghua University

     Tsinghua University

     China

     
  2. [2] National Taiwan University

     National Taiwan University

     Taiwán

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 15, Nº Extra 1 (Special volume), 2016, págs. 859-897
• Idioma: inglés
• Enlaces
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• Resumen

  • We study the following coupled Schrödinger 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.