Multi-bump solutions for Choquard equation with deepening potential well

2015/10/06 by Claudianor O. Alves, Alves, Claudianor O., Alânnio B. Nóbrega +3
Engineering · Mathematics · #35J20 #35J65 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1510.01409

openalex publication_date 2015/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the existence of multi-bump solutions to Choquard equation -Δu + (λa(x)+1)u=((1)/(|x|^μ)∗ |u|^p)|u|^p-2u in \R³, where μ∈ (0,3), p∈(2, 6-μ), λ is a positive parameter and the nonnegative function a(x) has a potential well Ω:=int (a^-1(0)) consisting of k disjoint bounded components Ω:=∪_j=1^kΩ_j. We prove that if the parameter λ is large enough then the equation has at least 2^k-1 multi-bump solutions.

Citations

• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• Nodal solutions for the Choquard equation
• On Gravity's role in Quantum State Reduction
• Existence of multi-bump solutions for a class of elliptic problems\n involving the biharmonic operator

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