Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two

TL;DR

This paper proves the existence of extremal functions for certain Trudinger-Moser inequalities in two dimensions, extending previous results to Riemannian surfaces and combining blow-up analysis with Carleson-Chang's results.

Contribution

It establishes the existence of extremal functions for Adimurthi-Druet type inequalities in dimension two, including Euclidean and Riemannian surface cases.

Findings

Existence of extremal functions proven for specific inequalities.

Extension of results to Riemannian surfaces.

Complementary to previous works by several authors.

Abstract

Combining Carleson-Chang's result with blow-up analysis, we prove existence of extremal functions for certain Trudinger-Moser inequalities in dimension two. This kind of inequality was originally proposed by Adimurthi and O. Druet, extended by the author to high dimensional case and Riemannian surface case, generalized by C. Tintarev to wider cases including singular form and by M. de Souza and J. M. do \'O to the whole Euclidean space R^2. In addition to the Euclidean case, we also consider the Riemannian surface case. The results in the current paper complement that of L. Carleson and A. Chang, M. Struwe, M. Flucher, K. Lin, and Adimurthi-Druet, our previous ones, and part of C. Tintarev.