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Boundary Behavior of Positive Solutions of a Semilinear Fractional Dirichlet Problem

Volume 3, Number 2 (2012), 75 - 90

Boundary Behavior of Positive Solutions of a Semilinear Fractional Dirichlet Problem

Communicated By: 
Enrique Zuazua
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Abstract

Let 0 < α < 2 and D be a bounded C1,1 domain in Rn,n ≥ 2. This paper deals with the existence and uniqueness of a positive continuous solution in D to the following fractional Dirichlet problem (−Δ)α2 u(x) = a(x)uσ(x), lim δ(x)→0 (δ(x))1−α2 u(x) = 0, where σ < 1, δ(x) = dist (x,∂D) and the function a is a positive measurable function in D satisfying some appropriate assumptions related to Karamata regular variation theory. We also investigate the boundary behavior of such solution.