Let $m,n\in\mathbb N$, let $U\subset\mathbb R^n$ be a bounded Lipschitz domain, and let $1<p<\infty$. Let $\mathcal A\subset W^{1,p}(U;\mathbb R^m)$ be nonempty and sequentially weakly closed in $W^{1,p}(U;\mathbb R^m)$. Let $r:=\min\{m,n\}$, and let
Assume also that $I$ is proper on $\mathcal A$, bounded below on $\mathcal A$, and coercive on $\mathcal A$ in the following sublevel-bounded sense: every sequence $(u_j)_{j=1}^{\infty}$ in $\mathcal A$ satisfying