On a singular semilinear elliptic boundary value problem and the boundary Harnack principle

Details On a singular semilinear elliptic boundary value problem and the boundary Harnack principle On a singular semilinear elliptic boundary value problem and the boundary Harnack principle

1999-06-09

arxiv.org/abs/math/9906213v1

Mathematics

Probability

Scientific paper

We consider the singular boundary-value problem \Delta u = f(u) in D; u|_dD= phi, where 1. D is a bounded C^2-domain of R^d, d >= 3 2. f: (0,1) -> (0,1) is a locally H\"older continuous function such that f(u) -> 1 as u -> 0 at the rate u^{-\alpha}, for some \alpha in (0,1), 3. and phi is a positive 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 D-bar. Such solutions are shown to satisfy a boundary Harnack principle. Probabilistic techniques are used in proving the main results.

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