Boundary blow up solutions for fractional elliptic equations

Abstract

In this article we study existence of boundary blow up solutions for some fractional elliptic equations including

(−Δ)^αu+u^p=f in Ω,

u=g on Ω^c,

lim _{x∈Ω,x→∂Ω}u(x)=∞,

where Ω is a bounded domain of class C², α∈(0,1) and the functions f :Ω→R and g :R^N\Ω⁻→R are continuous. We obtain existence of a solution u when the boundary value g blows up at the boundary and we get explosion rate for u under an additional assumption on the rate of explosion of g. Our results are extended for an ample class of elliptic fractional nonlinear operators of Isaacs type.