  We prove that for each positive integer $N$ the set of smooth, zero degree
maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three
properties:
  (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to
\mathbb{S}^2$ which agrees with $\psi$ on the boundary of the unit ball;
  (2) this map $u$ has at least $N$ singular points in $\mathbb{B}^3$;
  (3) the Lavrentiev gap phenomenon holds for $\psi$, i.e., the infimum of the
Dirichlet energies $E(w)$ of all smooth extensions $w\colon
\mathbb{B}^3\to\mathbb{S}^2$ of $\psi$ is strictly larger than the Dirichlet
energy $\int_{\mathbb{B}^3} |\nabla u|^2$ of the (irregular) minimizer $u$, is
dense in the set of all smooth zero degree maps $\phi\colon
\mathbb{S}^2\to\mathbb{S}^2$ endowed with the $W^{1,p}$-topology, where $1\le p
< 2$. This result is sharp: it fails in the $W^{1,2}$ topology on the set of
all smooth boundary data.
