Global Lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators

Abstract

This article is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this article is to establish some global estimates for nonuniformly elliptic equations in divergence form as−div(|∇u|p−2∇u+a(x)|∇u|q−2∇u)=−div(|F|p−2F+a(x)|F|q−2F), which arises from double-phase functional problems. In particular, the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators. This work extends that of Colombo and Mingione (2016) [23] and Byun and Oh (2017) [9] by dealing with the global estimates in Lorentz spaces. It also extends our recent result Tran and Nguyen (2020) [66] concerning new estimates of divergence elliptic equations using cutoff fractional maximal operators. For future research, the approach developed in this article will allow global estimates of distributional solutions to nonuniformly nonlinear elliptic equations to be obtained in the framework of other spaces.