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

In this paper we consider the semilinear elliptic problem u = 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 1 0 (1 − r)a(r)dr = +∞. For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.  2005 Elsevier Inc. All rights reserved