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

Abstract

We consider the equation −Δu=|x|α|u|p−1u for any α≥0, either in R2 or in the unit ball B of R2 centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as p→+∞ 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 α=0 and Dirichlet boundary conditions.