Nonexistence of extremals for an improved Adimurthi-Druet inequality involving Lp-norm on a closed Riemann surface

It is well known that the Adimurthi-Druet inequality admits extremal function, when the perturbation parameter is sufficiently small. As for the question of when extremal function does not exist, Mancini-Thizy first solved this problem by the method of energy estimate in (J. Differential Equations). After that Yang extended the work to a closed Riemann surface in (Sci. China Math.). In this paper, we generalize Yang’s result to a version involving L p -norms for any p > 1 . Moreover, this work complements our result in (Acta Math. Sin.) and extends Wang’s result (Commun. Pure Appl. Anal.) in Euclidean space.