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(x) |u|θ + χ{u̸=0} f(x) in Ω, with boundary conditions u = 0 on ∂Ω, where Ω is an open bounded subset of RN, 1 < p < N, -div(a(x, u,Du)) is a Leray-Lions operator defined on W1,p0 (Ω), a0 ∈ LN/p(Ω), a0 > 0, 0 < θ ≤ 1, χ{u̸=0} is a characteristic function, f ∈ LN/p(Ω) and H(x, s, ξ) is a Carath'eodory function such that -c0 a(x, s, ξ)ξ ≤ H(x, s, ξ) sign(s) ≤ γ a(x, s, ξ)ξ a.e. x ∈ Ω, ∀s ∈ R, ∀ξ ∈ RN. For ∥a0∥N/p and ∥f∥N/p sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that the function exp(δ|u|) - 1 belongs to W1,p0 (Ω) for some δ ≥ γ. This solution satisfies some a priori estimates in W1,p0 (Ω). Keywords: nonlinear problems, existence, singularity. [ABSTRACT FROM AUTHOR]