Analytical obstructions to the weak approximation of Sobolev mappings into manifolds

For any integer $p \geq 2$, 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 $s \in \mathbb{R}$, $s \geq 1$, $sp \in \mathbb{N}$, and $sp \ge 2$.