Quasilinear Riccati type equations with distributional data in Morrey space framework

In the framework of Morrey or Lorentz–Morrey spaces, we characterize the existence of solutions to the quasilinear Riccati type equation{−divA(x,∇u)=|∇u|q+σinΩ,u=0on∂Ω, with a distributional datum σ  . Here divA(x,∇u) is a quasilinear elliptic operator modelled after the p  -Laplacian, p>1p>1, but with a very general nonlinear structure, and Ω is a sufficiently flat domain in the sense of Reifenberg. The existence results are obtained in the natural or super-natural range of the gradient growth, i.e., q≥pq≥p.Motivated by the analysis of quasilinear Riccati type equation, a substantial part of the paper is also devoted to the Calderón–Zygmund type gradient regularity for the boundary value problem{divA(x,∇u)=div|f|p−2finΩ,u=0on∂Ω. We obtain regularity estimates in some weighted and unweighted function spaces as well as natural Lorentz–Morrey spaces associated to the Riccati type equation above.