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

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