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

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].