# Nonexistence of extremals for an inequality of Adimurthi-Druet on a closed Riemann surface

## TL;DR

This paper proves that for certain parameters, extremals do not exist for a specific inequality of Adimurthi-Druet on closed Riemann surfaces, extending previous results and clarifying the inequality's attainability conditions.

## Contribution

The paper establishes the nonexistence of extremals for the Adimurthi-Druet inequality on closed Riemann surfaces for a range of parameters, complementing earlier findings.

## Key findings

- Extremals do not exist for the inequality when \\alpha \\in (\\\alpha^*, \\\lambda_1(\\\Sigma))
- Identifies a threshold \\alpha^*<\\lambda_1(\\\Sigma) for nonattainability
- Extends previous results by Mancini-Thizy and the authors' earlier work.

## Abstract

Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(\Sigma,g)$. Precisely, if $\lambda_1(\Sigma)$ is the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition, then there exists a positive real number $\alpha^\ast<\lambda_1(\Sigma)$ such that for all $\alpha\in (\alpha^\ast,\lambda_1(\Sigma))$, the supremum $$\sup_{u\in W^{1,2}(\Sigma,g),\,\int_\Sigma udv_g=0,\,\|\nabla_gu\|_2\leq 1}\int_\Sigma \exp(4\pi u^2(1+\alpha\|u\|_2^2))dv_g$$ can not be attained by any $u\in W^{1,2}(\Sigma,g)$ with $\int_\Sigma udv_g=0$ and $\|\nabla_gu\|_2\leq 1$, where $W^{1,2}(\Sigma,g)$ denotes the usual Sobolev space and $\|\cdot\|_2=(\int_\Sigma|\cdot|^2dv_g)^{1/2}$ denotes the $L^2(\Sigma,g)$-norm. This complements our earlier result in [39].