On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

Abstract

On a bounded C 2-domain \(D \subset {\mathbb R}^d \) we consider the singular boundary-value problem 1/2Δu=f(u) in D, u D =φ, where d≥3, f:(0,∞)→(0,∞) is a locally Hölder continuous function such that f(u)→∞ as u→0 at the rate u −α, for some α∈(0,1), and φ is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in \(\bar D\). Such solutions are shown to satisfy a boundary Harnack principle.

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Athreya, S. On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle. Potential Analysis 17, 293–301 (2002). https://doi.org/10.1023/A:1016122901605

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  • semi-linear partial differential equations
  • boundary Harnack principle