Sharp interaction estimates and their application: existence of normalized ground states to coupled Schrödinger systems with potentials

2021/07/27 by Yinbin Deng, Deng, Yinbin, Qihan He +3
Mathematics · #35J20 #35J50 #35J61 #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.2107.12570

openalex publication_date 2021/07/27 · openalex created_date 2021/08/02 · openalex updated_date 2026/07/28

Abstract

In this paper, our aim is to prove the existence of normalized ground state for the following Schrödinger systems with potentials \begincases -Δu₁+V₁(x)u₁+λ₁ u₁=∂₁ G(u₁,u₂) \quadamp;\hboxin ℝ^N,
-Δu₂+V₂(x)u₂+λ₂ u₂=∂₂G(u₁,u₂) \quadamp;\hboxin ℝ^N,
0-∞, which are allowed to be singular at some points. And the nonlinearities G(u₁,u₂) are considered of the form \begincases G(u₁, u₂):=∑_i=1^ℓ(μ_i)/(p_i)|u₁|^p_i+∑_j=1^m(ν_j)/(q_j)|u₂|^q_j+∑_k=1ⁿβ_k |u₁|^r_1,k|u₂|^r_2,k,~~ℓ,m,n∈ ℕ⁺₀, μ_i, ν_j,β_k>0, ~21, i=1,2,⋯, ℓ; j=1,2,⋯, m; k=1,2,⋯, n. \endcases Under the mass sub-critical assumption, the normalized ground states are obtained as the minimum of the functional J on the manifold S_a1,a2. Since the functional is not weak lower semi-continuous, to prove the minimizing problem is achievable, the key step is establishing the strict sub-additive inequality. Among its main ingredients is the study of the sharp decay of the positive solutions and the interaction estimates.

Citations

• Uniqueness of positive solutions of Δu−u+up=0 in Rn
• A relation between pointwise convergence of functions and convergence of functionals
• 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

Related

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• Coupled nonlinear Schrödinger equations with point interaction: existence and asymptotic behaviour
• Normalized solutions for Schrödinger system with subcritical Sobolev exponent and combined nonlinearities
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• Ground states for a coupled nonlinear Schrödinger system