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

Abstract

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′. 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.