  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.
