A Discontinuous Solution of the Critical n-Laplace System with Antisymmetric Potential

Abstract: Let $n&gt;2$. We construct a map $U\in W<sup>{1,n}(B<sup>n,\mathbb{R}<sup>{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $Bⁿ \setminus {0}$, together with an antisymmetric potential $$Ω\in L<sup>n(B<sup>n,so(n+2)\otimes\mathbb{R}<sup>n)$$ such that $$-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=Ω\cdot |\nabla U|^{n-2}\nabla U$$ in $D&#39;(B<sup>n)$. This gives a negative answer to a regularity question posed by Rivière. Our potential admits the Lorentz-space regularity $Ω\in \bigcap_{q&gt;2}L^{(n,q)} \setminus L<sup>{(n,2)}$. In addition for given $1&lt;p&lt;\infty$ we can enforce $\nabla U \in L<sup>{(n,p)}$ but $\nabla U \notin L<sup>{(n,1)}$. The construction does not give a counterexample to regularity for weakly $n$-harmonic maps or for higher-dimensional $H$-systems. The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.