Nodal solutions for the Choquard equation

2015/03/31 by Marco Ghimenti, Jean Van Schaftingen · 217 citations
Mathematics · #Action (physics) #Advanced Mathematical Physics Problems #Compact space #Construct (python library) #Mathematical analysis #Mathematical optimization #Mathematics #Minimax #Nonlinear Partial Differential Equations #Nonlinear system #Numerical methods in inverse problems #Physics #Pure mathematics #Quantum mechanics #Riesz potential #Sign (mathematics) #math.AP #msc:35J20 #msc:35J91

paper · pdf · doi:10.1016/j.jfa.2016.04.019

published in Journal of Functional Analysis 271(1), 107-135 (Elsevier BV) · 23 pages, revised version with additional details and symmetry properties of odd solutions

arxiv created 2016/02/01 · openalex publication_date 2016/05/04 · arxiv updated 2017/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We consider the general Choquard equations -Δu + u = (I_α∗ |u|^p) |u|^{p - 2} u where I_α is a Riesz potential. We construct minimal action odd solutions for p ∈ ((N + α)/(N), (N + α)/(N - 2)) and minimal action nodal solutions for p ∈ (2,(N + α)/(N - 2)). We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais--Smale sequences. The nonlinear Schrödinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.

Citations

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

Cited by

• Normalized solutions to critical Choquard systems with linear and nonlinear couplings
• Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity
• On doubly nonlocal p-fractional coupled elliptic system
• Infinitely many non-radial solutions to a critical Choquard equation
• Positive solutions to fractional p-Laplacian Choquard equation on lattice graphs
• On the ground-state wave function of the one-dimensional polaron in the strong-coupling limit
• Existence and axial symmetry of minimal action odd solutions for 2-D Schrödinger-Newton equation
• Multi-bump solutions for Choquard equation with deepening potential well
• On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation
• Existence of nontrivial weak solutions for a quasilinear Choquard equation. [europepmc]

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• Nonlinear Choquard equations involving nonlocal operators