Asymptotic Estimates and Qualitatives Properties of an Elliptic Problem in Dimension Two

TL;DR

This paper investigates the asymptotic behavior and qualitative properties of positive solutions to a semilinear elliptic problem in two dimensions with large nonlinear exponents, establishing a limit problem and solution characteristics.

Contribution

It introduces new asymptotic estimates for solutions and characterizes their level sets and nondegeneracy, linking the original problem to a limit problem in two dimensions.

Findings

Asymptotic estimates for solutions as the exponent grows large

Characterization of solution level sets

Proof of nondegeneracy of solutions

Abstract

In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates which enable us to associate a "limit problem" to the initial one. Usong these estimates we prove some quantitative properties of the solution, namely characterization of level sets and nondegeneracy.