Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u⩾0 in Ω, with the boundary blow-up condition u|∂Ω=+∞, where Ω is a bounded domain in RN (N⩾2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f1) and (f2) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.