A Refinement of the First Eigenvalue and Eigenfunction of the Linearized Moser-Trudinger Problem

We revisit the following Moser-Trudinger problem {−Δu=λueu2in Ω,u>0in Ω,u=0on ∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} -\Delta u=\lambda ue^{u^{2}} &\text{in } \Omega , \\ u>0&\text{in } \Omega , \\ u=0 &\text{on } \partial \Omega , \end{cases} $$\end{document} where Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \subset \mathbb{R}^{2}$\end{document} is a smooth bounded domain and λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda >0$\end{document} is sufficiently small. Qualitative analysis of peaked solutions for Moser-Trudinger type equation in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{2}$\end{document} has been widely studied in recent decades. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions for the corresponding linearized Moser-Trudinger problem by using a variety of local Pohozaev identities combined with some elliptic theory in dimension two. Here we give some fine estimates for the first eigenvalue and eigenfunction of the linearized Moser-Trudinger problem. Since this problem is a critical exponent for dimension two and will lose compactness, we have to obtain some new and technical estimates.