The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient

In this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is −Δpu=β|∇u|q+f in Ω, u=0 on ∂Ω, where Ω is a bounded open subset of RN, N≥2, 1<p<N, Δpu=div|∇u|p−2∇u, p−1<q<p, β is a positive constant and f is a measurable function satisfying suitable summability conditions depending on q and a smallness condition.