Existence and multiplicity of positive solutions for classes of singular elliptic PDEs

We consider the boundary value problem−Δu=ϕg(u)u−αin Ω,u=0on ∂Ω, where Ω⊂RNΩ⊂RN is a bounded domain, ϕ   is a nonnegative function in L∞(Ω)L∞(Ω) such that ϕ>0ϕ>0 on some subset of Ω   of positive measure, and g:[0,∞)→Rg:[0,∞)→R is continuous. We establish the existence of three positive solutions when g(0)>0g(0)>0 (positone), the graph of sα+1g(s) is roughly S-shaped, and α>0α>0. We also prove that there exists at least one positive solution when g(0)<0g(0)<0 (semipositone), g(s)g(s) is eventually positive for s>0s>0, and 0<α<10<α<1. We employ the method of sub-super solutions to prove our results.