Singular elliptic problems: Existence, non-existence and boundary behavior

We deal with the existence, uniqueness and boundary behavior of positive solutions of the semilinear elliptic equation −Δu=ρa(x)g(u)+λb(x)f(u) in Ω−Δu=ρa(x)g(u)+λb(x)f(u) in Ω, under Dirichlet boundary conditions, where Ω⊂RNΩ⊂RN is a bounded domain with smooth boundary ∂Ω∂Ω, a,b,g,fa,b,g,f are continuous non-negative real valued functions. The main feature here is that either gg or ff (or both of them) are singular at 0 in the sense that g(t),f(t)⟶t→0∞ and ρ,λ≥0ρ,λ≥0 are parameters. Our results require no symmetry from either aa or bb and no monotonicity on ff or gg. Penalty arguments as well as variational principles are exploited.