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

TL;DR

This paper proves the existence of very weak solutions for nonlinear elliptic systems of p-Laplacian type with minimal regularity assumptions on the data, expanding the class of well-posed problems.

Contribution

It introduces a refined a priori estimate that establishes duality relations in weighted Lebesgue spaces, allowing solutions with less regular data.

Findings

Existence of solutions with divergence of q-integrable functions as data

Solutions are well-posed in larger function spaces than classical settings

Refined a priori estimates enable duality-based analysis

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.