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

We revisit the following Moser-Trudinger problem

$$ \textstyle\begin{cases} -\Delta u=\lambda ue^{u^{2}} &\text{in } \Omega , \\ u>0&\text{in } \Omega , \\ u=0 &\text{on } \partial \Omega , \end{cases} $$

where \(\Omega \subset \mathbb{R}^{2}\) is a smooth bounded domain and \(\lambda >0\) is sufficiently small. Qualitative analysis of peaked solutions for Moser-Trudinger type equation in \(\mathbb{R}^{2}\) 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.