Bubbling solutions for a planar exponential nonlinear elliptic equation with a singular source

2019/08/15 by Dong, Jingyi, Hu, Jiamei, Zhang, Yibin
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1908.05532

openalex publication_date 2019/08/15 · openalex created_date 2019/08/22 · openalex updated_date 2026/07/28

Abstract

Let Ω be a bounded domain in ℝ² with smooth boundary, we study the following elliptic Dirichlet problem \begincases -Δυ= e^υ-sϕ₁-4παδ_p-h(x) \textrmin Ω,
υ=0 \textrmon ∂Ω, \endcases where s>0 is a large parameter, h∈ C^0,γ(Ω), p∈Ω, α∈(-1,+∞)∖ℕ, δ_p denotes the Dirac measure supported at point p and ϕ₁ is a positive first eigenfunction of the problem -Δϕ=λϕ under Dirichlet boundary condition in Ω. If p is a strict local maximum point of ϕ₁, we show that such a problem has a family of solutions υ_s with arbitrary m bubbles accumulating to p, and the quantity ∫_Ωe^υ_s→8π(m+1+α)ϕ₁(p) as s→+∞.

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