  In this paper, we build infinitely many non-radial sign-changing solutions to
the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -\Delta
u&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }\Omega,\\ u&=0, &\hbox{ on
}\partial\Omega. \end{array}\right. \eqno(P) \end{equation*} on the annulus
$\Omega:=\{x\in \mathbb{R}^N: a<|x|<b\}$, $N\geq 3.$ In particular, for any
integer $k$ large enough, we build a non-radial solution which look like the
unique positive solution $u_0$ to $(P)$ crowned by $k$ negative bubbles
arranged on a regular polygon with radius $r_0$ such that
$r_0^{\frac{N-2}{2}}u_0(r_0)=:\displaystyle\max_{a\leq r\leq
b}r^{\frac{N-2}{2}}u_0(r).$
