Non-Regularizing properties of $n$-Laplace systems with antisymmetric potentials in critical Lebesgue spaces

For every $n\ge 3$ we construct a bounded discontinuous map $u\in W^{1,n}({\mathbb{B}}^n,{\mathbb{R}}^{n+2})$ solving an $n$-Laplace system with an antisymmetric potential $\mathrm{\Omega }\in L^n$. This shows that a recent regularity result on $n$-Laplace systems with antisymmetric potentials in Lorentz spaces by the authors is sharp in the sense that we cannot move from Lorentz spaces to classical Lebesgue spaces. This gives in particular a negative answer to a question by Rivière.