2015/10/13 by Phuoc‐Tai Nguyen, Phuoc-Tai Nguyen, Nguyen, Phuoc-Tai
Computer Science · Mathematics · #35J10 #35J60 #35J75 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35J10 #msc:35J60 #msc:35J75
paper · pdf · doi:10.48550/arxiv.1510.03803
27 pages
openalex publication_date 2015/10/13 · arxiv created 2015/10/27 · arxiv updated 2015/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be a smooth bounded domain in ℝN and δ(x)=dist (x,∂ Ω). Assume μ>0, ν is a nonnegative finite measure on ∂ Ω and g ∈ C(Ω× ℝ+). We study positive solutions of (P) -Δu - \fracμδ2 u = g(x,u) in Ω, tr^*(u)=ν. Here tr^*(u) denotes the normalized boundary trace of u which was recently introduced by M. Marcus and P. T. Nguyen. We focus on the case 0<μ< CH(Ω) (the Hardy constant for Ω) and provide some qualitative properties of solutions of (P). When g(x,u)=uq with q>1, we prove that there is a critical value q^* (depending only on N, μ) for (P) in the sense that if 1<q<q^* then (P) admits a solution under a smallness assumption on ν, but if q ≥ q^* this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical. We also investigate the case where the g is linear or sublinear and give some existence results for (P).