Symmetry breaking results for problems with exponential growth in the unit disk

TL;DR

The paper analyzes a two-dimensional variational problem with exponential growth in the unit disk, weighted by $|x|^\alpha$, focusing on the symmetry of maximizers. It derives an precise asymptotic description for radial maximizers as $\alpha\to\infty$, showing convergence to the first Laplacian eigenfunction under a suitable scaling, and obtains the leading order of the radial maximal value. The authors demonstrate symmetry breaking by proving that, for $\gamma$ near the critical threshold $4\pi$, nonradial maximizers outperform radial ones when $\alpha$ is large, and they provide a nonperturbative estimate giving a sharp bound on $\gamma$ beyond which radial symmetry cannot persist. The results connect to Hénon-type phenomena and extend symmetry-breaking analysis to exponential nonlinearities, offering insights into when radial minimizers fail to be optimal.

Abstract

We investigate some asymptotic properties of extrema to a two-dimensional variational problem in the unit disk. Some results about non-radialicity of solutions are given.

Theorems & Definitions (23)

• theorem 1: Smets, Su, Willem
• theorem 2
• theorem 3
• remark thmcounterremark
• proposition thmcounterproposition
• proof
• remark thmcounterremark
• lemma thmcounterlemma
• proof
• theorem 4