2015/11/10 by Alexander Grigorʼyan, Alexander Grigor'yan, Igor E. Verbitsky +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Divergence (linguistics) #Domain (mathematical analysis) #Elliptic curve #Elliptic operator #Function (biology) #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Operator (biology) #Pointwise #Pure mathematics #Riemannian manifold #Sign (mathematics) #math.AP #msc:35J61 #msc:58J05
paper · pdf · doi:10.1007/s11854-019-0004-z
published as J. Anal. Math. 137 (2019), no. 2, 559-601 · 25 pages
openalex created_date 2016/06/24 · arxiv created 2018/02/13 · openalex publication_date 2019/03/01 · arxiv updated 2020/11/10 · openalex updated_date 2026/08/05
We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type u(x) - ∫ΩG(x, y) g(u(y)) d σ(y) = h, where (Ω, σ) is a locally compact measure space, G(x, y)\colon Ω× Ω→ [0, +∞] is a kernel, h ≥ 0 is a measurable function, and g\colon [0, ∞)→ [0, ∞) is a monotone function. This problem is motivated by the semilinear fractional Laplace equation (-Δ)^\fracα2 u - g(u) σ= μ in Ω, u=0 in Ωc, with measure coefficients σ, μ, where g(u)=uq, q ∈ ℝ ∖\0\, and 0<α<n, in domains Ω⊆ℝn, or Riemannian manifolds, with positive Green's function G.