Positive normalized solutions to nonlinear elliptic systems in \R⁴ with critical Sobolev exponent

2021/07/19 by Xiao Luo, Xiaolong Yang, Luo, Xiao +3
Mathematics · #35B09 #35B33 #35B40 #35J50 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2107.08708

openalex publication_date 2021/07/19 · openalex created_date 2021/08/02 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begincases -Δu+λ₁u=μ₁u³+α₁|u|^p-2u+βv²u\quadamp;\hboxin~\R⁴,
-Δv+λ₂v=μ₂v³+α₂|v|^p-2v+βu²v\quadamp;\hboxin~\R⁴,
\endcases under the mass constraint ∫_\R⁴u²=a₁²\quadand ∫_\R⁴v²=a₂², where a₁,a₂ are prescribed, μ₁,μ₂,β>0; α₁,α₂∈ \R, p ∈ (2,4) and λ₁,λ₂ ∈ \R appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., α_i<0(i=1,2). Then turning to the case of α_i>0 (i=1,2), if 2

Citations

• Uniqueness of positive solutions of Δu−u+up=0 in Rn
• Nonlinear scalar field equations, II existence of infinitely many solutions
• Nonlinear Schr�dinger equations and sharp interpolation estimates
• A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems
• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
• Normalized ground states for the NLS equation with combined nonlinearities
• Ground states for 3D dipolar Bose-Einstein condensate involving quantum fluctuations and three-body losses

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