A Singular Nonlinear Problems With Natural Growth in the Gradient

In this paper, we consider the equation -div\, (a (x, u, Du) =H (x, u, Du) \\+a₀ (ₗ) u ^{}+\ₔ ₀\\, f (x) in, with boundary conditions u=0 on, where is an open bounded subset of R^N, 1 0, 0< 1, \ₔ ₀\ is a characteristic function, f L^N/p () and H (x, s, ) is a Carath\'eodory function such that -c₀\, a (x, s, ) \, H (x, s, ) \, sign (s) \, a (x, s, ) a. e. x, s\, \, , R^N. For a₀₍/₏ and f₍/₏ sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that the function (u) -1 belongs to W₀^1, p () for some. This solution satisfies some a priori estimates in W₀^1, p ().