Elliptic singular problems with a quadratic gradient term

Abstract

We deal with existence and nonexistence of positive classical solutions to the Dirichlet problem for the quasilinear singular elliptic equation −∆u = λ β(u) |∇u| 2 + Ψ(x) in Ω, where Ω ⊂ R N (N ≥ 3) is a domain with smooth boundary ∂Ω, λ > 0 is a real parameter, β : (0, ∞) → (0, ∞) s→0 is a C 1 -function, possibly singular at zero in the sense that β(s) → ∞, and Ψ : Ω → [0, ∞) is continuous. No monotonicity condition whatsoever is imposed upon β.