Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems

We consider the equation $-\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ for any $\alpha \ge 0$, 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 $p\to +\mathrm{\infty }$ 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.