  We consider the equation $-\Delta u= |x|^{\alpha}|u|^{p-1}u$ for any
$\alpha\geq 0$, either in $\mathbb R^2$ or in the unit ball $B$ of $\mathbb
R^2$ centered at the origin with Dirichlet or Neumann boundary conditions. We
give a sharp description of the asymptotic behavior as $p\rightarrow +\infty$
of all the radial solutions to these problems. In particular, we show that
there is no uniform a priori bound (in $p$) for nodal solutions under Neumann
or Dirichlet boundary conditions. This contrasts with the recently shown fact
that positive solutions have uniform a priori bounds for $\alpha=0$ and
Dirichlet boundary conditions.
