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 $alphain (alpha^ast,lambda_1(Sigma))$, the supremum $$sup_{uin W^{1,2}(Sigma,g),,int_Sigma udv_g=0,,| abla_gu|_2leq 1}int_Sigma exp(4pi u^2(1+alpha|u|_2^2))dv_g$$ can not be attained by any $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $| abla_gu|_2leq 1$, 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].