Solutions to nonlinear elliptic equations with a gradient

In this article, we consider existence and nonexistence of solutions to problem equation(0.1){-Δpu+g(x,u)|∇u|p=finΩ,u=0on∂Ω with 1 < p < ∞, where f   is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in ℝN(N≥2)ℝN(N≥2). Especially, under the condition that g(x,s)=1/|sα|(α>0)g(x,s)=1/|s|α(α>0) is singular at s = 0, we obtain that α < p   is necessary and sufficient for the existence of solutions in W01,p(Ω) to problem (0.1) when f is sufficiently regular.