  We study the self-similar solutions of the equation \[ u_{t}-div(| \nabla u|
^{p-2}\nabla u)=0, \] in $\mathbb{R}^{N},$ when $p>2.$ We make a complete study
of the existence and possible uniqueness of solutions of the form \[
u(x,t)=(\pm t)^{-\alpha/\beta}w((\pm t)^{-1/\beta}| x|) \] of any sign, regular
or singular at $x=0.$ Among them we find solutions with an expanding compact
support or a shrinking hole (for $t>0),$ or a spreading compact support or a
focussing hole (for $t<0).$ When $t<0,$ we show the existence of positive
solutions oscillating around the particular solution $U(x,t)=C_{N,p}(| x|
^{p}/(-t))^{1/(p-2)}.$
