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Singular semilinear elliptic inequalities in the exterior of a compact set

Published online by Cambridge University Press:  22 May 2013

Marius Ghergu
Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland ([email protected])
Steven D. Taliaferro
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA ([email protected])

Abstract

We study the semilinear elliptic inequality –Δu ≥ φ(δK (x))f(u) in ℝN / K, where φ, f are positive and non-increasing continuous functions. Here K ⊂ ℝN (N ≥ 3) is a compact set with finitely many components, each of which is either the closure of a C2 domain or an isolated point, and δK (x) = dist(x, ∂K). We obtain optimal conditions in terms of φ and f for the existence of C2-positive solutions. Under these conditions we prove the existence of a minimal solution and we investigate its behaviour around ∂K as well as the removability of the (possible) isolated singularities.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2013 

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