  The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and
$p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a
modified form with $\lambda(1+u)^p$ as right-hand-side with an additional
eigenvalue parameter $\lambda>0$ in the unit ball, in the latter case together
with Dirichlet boundary conditions. As for entire regular radial solutions we
prove oscillatory behaviour around the explicitly known radial {\it singular}
solution, provided $p\in((n+4)/(n-4),p_c)$, where $p_c\in ((n+4)/(n-4),\infty]$
is a further critical exponent, which was introduced in a recent work by
Gazzola and the second author. The third author proved already that these
oscillations do not occur in the complementing case, where $p\ge p_c$.
  Concerning the Dirichlet problem we prove existence of at least one singular
solution with corresponding eigenvalue parameter. Moreover, for the extremal
solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue
problem, we prove smoothness as long as $p\in((n+4)/(n-4),p_c)$.
