Existence and regularity results for some fully nonlinear singular or degenerate equation

International audience ; In this article we prove existence, uniqueness and regularity for the singular equation |∇u| α (F (D 2 u) + h(x) • ∇u) + c(x)|u| α u + p(x)u −γ = 0 in Ω u > 0 in Ω, u = 0 on ∂Ω when p is some continuous and positive function, c and h are continuous, α > −1 and F is Fully non linear elliptic. Some conditions on the first eigenvalue for the operator −|∇u| α (F (D 2 u) + h(x) • ∇u) − c(x)|u| α u are required. The results generalize the well known results of Lazer and McKenna.