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

作者：Pan, Kefan; Yang, Jing^*

来源：Acta Applicandae Mathematicae, 2023, 187(1): 11.

DOI：10.1007/s10440-023-00603-4

• The linearized Moser-Trudinger problem
• The first eigenvalue
• The first eigenfunction
• Asymptotic behavior

We revisit the following Moser-Trudinger problem @@@ .{-Delta u = lambda ue(u2) in Omega @@@ {u > 0 in Omega @@@ {u = 0 on partial derivative Omega @@@ where Omega. R-2 is a smooth bounded domain and lambda > 0 is sufficiently small. Qualitative analysis of peaked solutions for Moser-Trudinger type equation in 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.

• 单位
  江苏科技大学