Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

2020/06/25 by Houwang Li, Wenming Zou, Li, Houwang +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2006.14387

openalex publication_date 2020/06/25 · openalex created_date 2020/07/02 · openalex updated_date 2026/07/28

Abstract

In the present paper, we study the normalized solutions with least energy to the following system: \begincases -Δu+λ₁u=μ₁ |u|^p-2u+βr₁|u|^r₁-2|v|^r₂u amp;\hboxin \mathbb R^N,
-Δv+λ₂v=μ₂ |v|^q-2v+βr₂|u|^r₁|v|^r₂-2v\quadamp;\hboxin \mathbb R^N,
∫_{\mathbb R^N}u²=a₁² \hboxand ∫_{\mathbb R^N}v²=a₂², \endcases where p,q,r₁+r₂ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on a₁,a₂ and β, we obtain the existence of the positive normalized ground state solution to the above system. We have solved some unsolved open problems in this area.

Citations

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

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