In this paper we consider the elliptic boundary blow-up problem { Δ u = ( a + ( x ) − ε a − ( x ) ) u p in Ω , u = ∞ on ∂ Ω where Ω is a bounded smooth domain of R N , a + , a − are positive continuous functions supported in disjoint subdomains Ω + , Ω − of Ω, respectively, p > 1 and ε > 0 is a parameter. We show that there exists ε ⁎ > 0 such that no positive solutions exist when ε > ε ⁎ , while a minimal positive solution exists for every ε ∈ ( 0 , ε ⁎ ) . Under the additional hypotheses that Ω ¯ + and Ω ¯ − intersect along a smooth ( N − 1 ) -dimensional manifold Γ and a + , a − have a convenient decay near Γ, we show that a second positive solution exists for every ε ∈ ( 0 , ε ⁎ ) if p < N ⁎ = ( N + 2 ) / ( N − 2 ) . Our proofs are mainly based on continuation methods.