Qualitative analysis for Moser-Trudinger nonlinearities with a low energy

TL;DR

This paper provides a detailed qualitative analysis of positive solutions to the Moser-Trudinger problem with low energy in two-dimensional domains, addressing issues like asymptotic behavior, uniqueness, and symmetry.

Contribution

It introduces new methods using local Pohozaev identities and ODE theory to analyze low-energy solutions of the critical Moser-Trudinger problem in 2D.

Findings

Characterization of asymptotic behavior of solutions

Results on uniqueness and symmetry of solutions

Analysis of Morse index and non-degeneracy

Abstract

We are concerned with the Moser-Trudinger problem \begin{equation*} \begin{cases} -\Delta u=\lambda ue^{u^2}~~&\mbox{in}~\Omega,\\[0.5mm] u>0 ~~ &{\text{in}~\Omega},\\[0.5mm] u=0~~&\mbox{on}~\partial \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^2$ is a smooth bounded domain and $\lambda>0$ is sufficiently small. Qualitative analysis for Moser-Trudinger nonlinearities has been studied in recent decades, however there is still a lot of clarity about this issue, even for a low energy. The reason is that this problem is a critical exponent for dimension two and will lose compactness. Here by using a variety of local Pohozaev identities, we qualitatively analyze the positive solutions of Moser-Trudinger problem with a low energy, which contains the Morse index, non-degeneracy, asymptotic behavior, uniqueness and symmetry of solutions. Since the fundamental solution of $-\Delta$ in $\Omega \subset \mathbb{R}^2$ is in logarithmic form and the corresponding bubble is exponential growth, more precise asymptotic behavior of the solutions is needed, which is of independent interest. Moreover, to obtain our results, some ODE's theory will be used to a prior estimate of the solutions and some elliptic theory in dimension two will play a crucial role.