Normalized multi-bump solutions for Choquard equation involving sublinear case

2025/05/09 by He Zhang, Zhang, He, Haibo Chen +2 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.06097

openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the existence of normalized multi-bump solutions for the following Choquard equation -ε²Δu +λu=ε^-(N-μ)(∫_ℝ^N(Q(y)|u(y)|^p)/(|x-y|^μ)dy)Q(x)|u|^p-2u, in ℝ^N, where N≥3, μ∈ (0,N), ε>0 is a small parameter and λ∈ℝ appears as a Lagrange multiplier. By developing a new variational approach, we show that the problem has a family of normalized multi-bump solutions focused on the isolated part of the local maximum of the potential Q(x) for sufficiently small ε>0. The asymptotic behavior of the solutions as ε→0 are also explored. It is worth noting that our results encompass the sublinear case p<2, which complements some of the previous works.

Citations

• Normalized concentrating solutions to nonlinear elliptic problems
• Semiclassical states for Choquard type equations with critical growth: critical frequency case
• Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well
• Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity
• Mass minimizers and concentration for nonlinear Choquard equations in \R^N
• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• On Gravity's role in Quantum State Reduction
• GRAVITATIONAL SELF-ENERGY AS THE LITMUS OF REALITY
• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

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• On the existence of normalized solutions to a class of fractional Choquard equation with potentials

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