Analytical obstructions to the weak approximation of Sobolev mappings into manifolds

For any integer $p\ge 2$, we construct a compact Riemannian manifold $\mathcal{N}$ such that if $\dim \mathcal{M}>p$, there is a map in the Sobolev space of mappings $W^{1,p}(\mathcal{M},\mathcal{N})$ which is not a weak limit of smooth maps into $\mathcal{N}$ due to a mechanism of analytical obstruction. For $p=4n-1$, the target manifold can be taken to be the sphere ${\mathbb{S}}^{2n}$ thanks to the construction by Whitehead product of maps with nontrivial Hopf invariant, generalizing the result by Bethuel for $p=4n-1=3$. The results extend to higher order Sobolev spaces $W^{s,p}$, with $s\in \mathbb{R}$, $s\ge 1$, $sp\in \mathbb{N}$, and $sp\ge 2$.