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

TL;DR

This paper analyzes the precise asymptotic behavior of all radial solutions to a class of planar semilinear elliptic equations with weight, revealing the lack of uniform bounds for nodal solutions as the nonlinearity grows.

Contribution

It provides a sharp description of the asymptotic behavior of radial solutions for weighted semilinear elliptic problems in two dimensions, including the first results on the unboundedness of nodal solutions.

Findings

No uniform a priori bounds for nodal solutions as p→∞

Positive solutions have uniform bounds when α=0 and Dirichlet conditions

Asymptotic behavior characterized for all radial solutions

Abstract

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.