Sharp boundary behaviour of solutions to semilinear nonlocal elliptic\n equations

2017/10/07 by Matteo Bonforte, Alessio Figalli, Bonforte, Matteo +2 · 2 citations
Computer Science · Mathematics · #35B45 #35B65 #35J61 #35K67 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1710.02731

openalex publication_date 2017/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate quantitative properties of nonnegative solutions u(x)\≥ 0\nto the semilinear diffusion equation \L u= f(u), posed in a bounded\ndomain \Ω\⊂ mathbb R^N with appropriate homogeneous Dirichlet or\nouter boundary conditions. The operator \L may belong to a quite\ngeneral class of linear operators that include the standard Laplacian, the two\nmost common definitions of the fractional Laplacian (-\Δ)^s (0<s<1) in\na bounded domain with zero Dirichlet conditions, and a number of other nonlocal\nversions. The nonlinearity f is increasing and looks like a power function\nf(u)\∼ u^p, with p\≤ 1.\n The aim of this paper is to show sharp quantitative boundary estimates based\non a new iteration process. We also prove that, in the interior, solutions are\nH "older continuous and even classical (when the operator allows for it). In\naddition, we get H "older continuity up to the boundary.\n Particularly interesting is the behaviour of solution when the number\n\(2s)/(1-p) goes below the exponent \γ \∈(0,1] corresponding to the\nH "older regularity of the first eigenfunction \L\Φ₁=\λ₁\n\Φ₁. Indeed a change of boundary regularity happens in the different\nregimes \(2s)/(1-p) gtreqqless \γ, and in particular a logarithmic\ncorrection appears in the "critical" case \(2s)/(1-p) = \γ. Indeed a\nchange of boundary regularity happens in the different regimes \(2s)/(1-p)\n gtreqqless \γ, and in particular a logarithmic correction appears in the\n"critical" case \(2s)/(1-p) = \γ. For instance, in the case of the\nspectral fractional Laplacian, this surprising boundary behaviour appears in\nthe range 0<s\≤ \(1-p)/(2).\n

Cited by

• Large Solutions for Fractional Laplacian on Infinite Cylindrical Domains
• Nonlocal degenerate parabolic hyperbolic equations on bounded domains. Part II: Existence

Related

• Singular boundary behaviour and large solutions for fractional elliptic equations
• Semilinear nonlocal elliptic equations with source term and measure data
• Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian
• A large class of nonlocal elliptic equations with singular nonlinearities
• The Dirichlet Problem for the fractional p-Laplacian evolution equation