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

We consider Lane-Emden and Nenon equations either in the plane or in the unit planar ball B centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior of all the radial solutions to these problems and we show that there is no uniform a priori bound for nodal solutions under Neumann or Dirichlet boundary conditions.