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{Δu=(a+(x)−εa−(x))upin Ω,u=∞on ∂Ω where Ω   is a bounded smooth domain of RNRN, a+a+, a−a− are positive continuous functions supported in disjoint subdomains Ω+Ω+, Ω−Ω− of Ω  , respectively, p>1p>1 and ε>0ε>0 is a parameter. We show that there exists ε⁎>0ε⁎>0 such that no positive solutions exist when ε>ε⁎ε>ε⁎, while a minimal positive solution exists for every ε∈(0,ε⁎)ε∈(0,ε⁎). Under the additional hypotheses that Ω¯+ and Ω¯− intersect along a smooth (N−1)(N−1)-dimensional manifold Γ   and a+a+, a−a− have a convenient decay near Γ  , we show that a second positive solution exists for every ε∈(0,ε⁎)ε∈(0,ε⁎) if p