Normalized ground states to a cooperative system of Schrödinger equations with generic L²-subcritical or L²-critical nonlinearity

2021/01/08 by Jacopo Schino, Schino, Jacopo · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2101.03076

openalex publication_date 2021/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We look for ground state solutions to the Schrödinger-type system \begincases -Δu_j + λ_j u_j = ∂_jF(u)
∫_\rn u_j² dx = a_j²
(λ_j,u_j) ∈ ℝ × H¹(ℝ^N) \endcases j ∈ \1,…,M\ with N,M≥1, where a=(a₁,…,a_M) ∈ ]0,∞[^M is prescribed and (λ,u) = (λ₁,…,λ_M,u₁,… u_M) is the unknown. We provide generic assumptions about the nonlinearity F which correspond to the L²-subcritical and L²-critical cases, i.e., when the energy is bounded from below for all or some values of a. Making use of a recent idea, we minimize the energy over the constraint \Set|u_j|_L²≤ a_j for all j and then provide further assumptions that ensure |u_j|_L²=a_j.

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• Normalized Solutions for the (2,q)-Laplacian Operator Between Mass-Critical Exponents
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