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

Based on a recent work of Mancini and Thizy(2019),we obtain the nonexistence of extremals for an inequality of Adimurthi and Druet(2004)on a closed Riemann surface(Σ,g).Precisely,ifλ1(Σ)is the first eigenvalue of the Laphace-Beltrami operator with respect to the zero mean value condition,then there exists a positive real numberα*<λ1(Σ)such that for allα∈(α*,λ1(Σ)),the supremum■cannot be attained by any u∈W1,2(Σ,g)with∫Σudvg=0 and‖▽gu‖2≤1,where W1,2(∑,g)denotes the usual Sobolev space and‖·‖2=(∫Σ|·|2 dvg)1/2denotes the L2(Σ,g)-norm.This complements our earlier result in Yang(2007).