2024/06/01 by Konstantinos T. Gkikas, Gkikas, Konstantinos T., Miltiadis Paschalis +1
Mathematics · #35J10 #35J25 #35J60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2406.00354
openalex publication_date 2024/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω⊂ℝN (N≥ 3) be a bounded C2 domain and Σ⊂∂Ω be a compact C2 submanifold of dimension k. Denote the distance from Σ by dΣ. In this paper, we study positive solutions of the equation (*) -Δu -μu/dΣ2 = g(u,|∇ u|) in Ω, where μ≤ ( (N-k)/(2) )2 and the source term g:ℝ×ℝ+ → ℝ+ is continuous and non-decreasing in its arguments with g(0,0)=0. In particular, we prove the existence of solutions of (*) with boundary measure data u=ν in two main cases, provided that the total mass of ν is small. In the first case g satisfies some subcriticality conditions that always ensure the existence of solutions. In the second case we examine power type nonlinearity g(u,|∇ u|) = |u|p|∇ u|q, where the problem may not possess a solution for exponents in the supercritical range. Nevertheless we obtain criteria for existence under the assumption that ν is absolutely continuous with respect to some appropriate capacity or the Bessel capacity of Σ, or under other equivalent conditions.