Global Lorentz estimates for non-uniformly nonlinear elliptic equations via fractional maximal operators

Abstract

This paper is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this paper is to establish some global estimates for non-uniformly elliptic in divergence form as follows \begin{align*} -\mathrm{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u) = - \mathrm{div}(|\mathbf{F}|^{p-2}\mathbf{F} + a(x)|\mathbf{F}|^{q-2}\mathbf{F}), \end{align*} that 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 \cite{CoMin2016,Byun2017Cava} by dealing with the global estimates in Lorentz spaces. This work also extends our recent result in \cite{PNJDE}, which is devoted to the new estimates of divergence elliptic equations using cut-off fractional maximal operators. For future research, the approach developed in this paper allows to attain global estimates of distributional solutions to non-uniformly nonlinear elliptic equations in the framework of other spaces.