We consider the elliptic boundary blow-up problem
where is a smooth bounded domain of , , are positive continuous functions supported in disjoint subdomains , of , respectively, and is a parameter. We show that there exists such that no positive solutions exist when , while a minimal positive solution exists for every . Under the additional hypotheses that and intersect along a smooth -dimensional manifold and , have a convenient decay near , we show that a second positive solution exists for every if . Our proofs are mainly based on continuation methods.