Existence of very weak solutions to elliptic systems of p-Laplacian type

We study vector valued solutions to non-linear elliptic partial differential equations with p-growth. Existence of a solution is shown in case the right hand side is the divergence of a function which is only q integrable, where q is strictly below but close to the duality exponent p′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p'$$\end{document}. It implies that possibly degenerate operators of p-Laplacian type are well posed in a larger class then the natural space of existence. The key novelty here is a refined a priori estimate, that recovers a duality relation between the right hand side and the solution in terms of weighted Lebesgue spaces.