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 $p^{\prime }$. 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.

References in corpus (1)

• Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems

Cited by in corpus (5)

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• Well posedness of nonlinear parabolic systems beyond duality
• On global $L^q$ estimates for systems with $p$-growth in rough domains
• A note on local $W^{1,p}$-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts
• An existence result for nonhomogeneous quasilinear parabolic equations beyond the duality pairing