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

2015/08/07 by Simão Correia, Correia, Simão, Filipe Oliveira +3 · 1 citation
Mathematics · #35B08 #35B09 #35J47 #35J50 (Primary) #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1508.01783

openalex publication_date 2015/08/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this work we consider the weakly coupled Schrödinger cubic system \begincases -Δu_i+λ_i u_i= μ_i u_i³+ u_i∑_{j≠ i}b_ij u_j²
u_i∈ H¹(ℝ^N;ℝ), i=1,…, d, \endcases where 1≤ N≤ 3, λ_i,μ_i >0 and b_ij=b_ji>0 for i≠ j. This system admits semitrivial solutions, that is solutions u=(u₁,…, u_d) with null components. We provide optimal qualitative conditions on the parameters λ_i,μ_i and b_ij 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 λ_i≡ λ and b_ij≡ b. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the d=2 case.

Citations

• Uniqueness of positive solutions of Δu−u+up=0 in Rn

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• Existence results for variational quasilinear elliptic systems involving the vectorial p-Laplacian

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