L^∞ Estimate for Some Nonlinear Elliptic Partial Differential Equations and Application to an Existence Result

Consider the nonlinear elliptic equation (E):\, A(u) + H(x,u,Du) = f(x) - div\,g(x) where A(u) = - div\,(a(x,u,Du)) + a₀ (x,u,Du) is a Leray–Lions operator defined on W₀1,p (Ω ) with a₀ (x,s,ξ )s α ₀ |s|ᵖ, α ₀ > 0, and where H is a first-order term satisfying |H(x,s,ξ )| C₀ + C₁ |ξ |ᵖ. The main goal of this paper is to prove an L^∞ estimate for the bounded solutions of $(E)$ when f belongs to Lq (Ω ) and g belongs to (Lʳ (Ω ))N with $r = p'q$ and max (1,N / p) < q + ∞. In view of the method and results developed in the author’s previous work, this implies the existence of a solution for equation $(E)$.