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(x) u θ+χ_≠ 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 W₀1,p(Ω), a₀∈ LN/p(Ω ), a₀> 0, 0<θ≤ 1, χ_≠ 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\,\, , ∀ξ ∈ RN. For a₀N/p and fN/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 W₀1,p(Ω) for some δ≥ γ. This solution satisfies some a priori estimates in W₀1,p(Ω).