Non-classical solutions of the $p$-Laplace equation

Abstract

In this paper we settle Iwaniec and Sbordone’s 1994 conjecture concerning very weak solutions to the $p$-Laplace equation. Namely, on the one hand we show that distributional solutions of the $p$-Laplace equation in $W^{1,r}$ for $p\ne 2$ and $r>\max \{1,p-1\}$ are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus disproving Iwaniec and Sbordone’s conjecture in general.