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

We consider the equation -Delta u = vertical bar x vertical bar(alpha vertical bar)u vertical bar(p-1)u for any alpha >= 0, either in R-2 or in the unit ball B of R-2 centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as p -> +infinity 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. This contrasts with the existence of uniform bounds for positive solutions, as shown in [32] for alpha = 0 and Dirichlet boundary conditions. (C) 2021 Published by Elsevier Inc.