Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with d ≥ 3 equations

Simão Correia, Filipe Oliveira, Hugo Tavares

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

In this work we consider the weakly coupled Schrödinger cubic system{−Δuiiuiiui 3+ui∑j≠ibijuj 2ui∈H1(RN;R),i=1,…,d, where 1≤N≤3, λii>0 and bij=bji>0 for i≠j. This system admits semitrivial solutions, that is solutions u=(u1,…,ud) with null components. We provide optimal qualitative conditions on the parameters λi, μi and bij under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the d=2 equations case. For d≥3 equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case λi≡λ and bij≡b. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the d=2 case.

Original languageEnglish
Pages (from-to)2247-2273
Number of pages27
JournalJournal Of Functional Analysis
Volume271
Issue number8
DOIs
Publication statusPublished - 15 Oct 2016

Keywords

  • Cubic Schrödinger systems of cooperative type
  • Gradient elliptic systems
  • Ground states
  • Semitrivial and fully nontrivial solutions

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