Multiplicity of positive solutions to boundary blow-up elliptic problems with sign-changing weights

In this paper we consider the elliptic boundary blow-up problem u = a+(x)− εa−(x) up in Ω, u=∞ on ∂Ω where Ω is a bounded smooth domain of RN, 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 thatΩ+ andΩ− 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