2020/03/08 by Yibin Zhang, 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.2003.04718
openalex publication_date 2020/03/08 · openalex created_date 2020/03/23 · openalex updated_date 2026/07/28
Let Ω be a bounded domain in ℝ2 with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-Hénon weight \begincases -∇(a(x)∇ u)+a(x)u=a(x)|x-q|2αup, ugt;0 \textrmin Ω,
(∂ u)/(∂ν)=0 \textrmon ∂Ω, \endcases where ν denotes the outer unit normal vector to ∂Ω, q∈Ω, α∈(-1,+∞)∖ℕ, p>1 is a large exponent and a(x) is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient a(x) and singular source q on the existence of concentrating solutions. We show that if q∈Ω is a strict local maximum point of a(x), there exists a family of positive solutions with arbitrarily many interior spikes accumulating to q; while if q∈∂Ω is a strict local maximum point of a(x) and satisfies ⟨∇ a(q), ν(q)⟩=0, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to q. In particular, we find that concentration at singular source q is always possible whether q∈Ω is an isolated local maximum point of a(x) or not.