Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics

2012/05/31 by Vitaly Moroz, Jean Van Schaftingen · 883 citations
Mathematics · #Advanced Mathematical Physics Problems #Elliptic curve #Geometry #Infinity #Mathematical analysis #Mathematical physics #Mathematics #Monotone polygon #Nonlinear Partial Differential Equations #Nonlinear system #Numerical methods in inverse problems #Physics #Quantum mechanics #Riesz potential #math.AP #msc:35B09 #msc:35B33 #msc:35B40 #msc:35J61 #msc:35Q55 #msc:45K05

paper · pdf · doi:10.1016/j.jfa.2013.04.007

published in Journal of Functional Analysis 265(2), 153-184 (Elsevier BV) · 23 pages, updated bibliography

arxiv created 2013/04/22 · openalex publication_date 2013/04/28 · arxiv updated 2013/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We consider a semilinear elliptic problem [- Δu + u = (I_α∗ \absu^p) \absu^{p - 2} u \quadin (ℝ^N),] where (I_α) is a Riesz potential and (p>1). This family of equations includes the Choquard or nonlinear Schrödinger-Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.

Citations

• Nonlinear Schr�dinger equations and sharp interpolation estimates

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