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

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• 作者：Simã ; o Correia^a ; ^{sfcorreia@fc.ul.pt" class="auth_mail" title="E-mail the corresponding author} ; Filipe Oliveira^b ; ^{fso@fct.unl.pt" class="auth_mail" title="E-mail the corresponding author} ; Hugo Tavares^c ; ^{htavares@math.ist.utl.pt" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35J47 ; 35J50 ; secondary ; 35B08 ; 35B09 ; 35Q55
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：271
• 期：8
• 页码：2247-2273
• 全文大小：464 K

文摘

In this work we consider the weakly coupled Schrödinger cubic system where 1≤N≤3$1\le N\le 3$, λ_i,μ_i>0${\lambda }_i,{\mu }_i>0$ and b_ij=b_ji>0$b_{ij}=b_{ji}>0$ for i≠j$i\ne j$. This system admits semitrivial solutions, that is solutions u=(u₁,…,u_d)$\mathbf{u}=(u_1,\dots ,u_d)$ with null components. We provide optimal qualitative conditions on the parameters λ_i${\lambda }_i$, μ_i${\mu }_i$ and b_ij$b_{ij}$ under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial.

This question had been clarified only in the 35bd3dee8abe5e1dee9966" title="Click to view the MathML source">d=2$d=2$ equations case. For d≥3$d\ge 3$ 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 35b446b747a" title="Click to view the MathML source">λ_i≡λ${\lambda }_i\equiv \lambda$ and 351adf6fb700d74" title="Click to view the MathML source">b_ij≡b$b_{ij}\equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the 35bd3dee8abe5e1dee9966" title="Click to view the MathML source">d=2$d=2$ case.