Existence of normalized solutions of a Hartree-Fock system with mass subcritical growth

2024/05/02 by Hua Jin, Y. Austin Chang, Jin, Hua +5
Mathematics · #35J50 #35R09 #37K45 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2405.01036

openalex publication_date 2024/05/02 · openalex created_date 2024/05/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we are concerned with normalized solutions in H_r¹(ℝ³) × H_r¹(ℝ³) for Hartree-Fock type systems with the form \be\lab Hartree-Fock \ -Δu +αϕ_u,v u=λ₁ u+ | u | ^2q-2 u+β | v | ^q | u | ^q-2 u ,
-Δv +αϕ_u,v v=λ₂ v+ | v | ^2q-2 v+β | u | ^q | v | ^q-2 v ,
∫_ℝ³ | u | ² \rm dx=a₁ , ∫_ℝ³ | v | ² \rm dx=a₂ , where ϕ_{u, v}(x):=∫_ℝ³ \fracu²(y)+v²(y)|x-y| \rm dy ∈ D^1,2(ℝ³). Here α,β>0, a₁,a₂>0 and 10 when 10 small when (4)/(3)≤ q < (3)/(2). The nonexistence of normalized solutions is also considered for (3)/(2)≤ q < (5)/(3). Also, the orbital stability of standing waves is obtained under local well-posedness assumptions of the evolution problem.

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