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

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 $$\underset{u\in W^{1,2}(\Sigma ,g),{\int }_{\Sigma }ud{v}_g=0,\Vert {\nabla }_{g}u{\Vert }_2\le 1}{\sup }{\int }_{\Sigma }\exp (4\pi u^2(1+\alpha \Vert u{\Vert }_2^2))d{v}_g$$ can not be attained by any $u\in W^{1,2}(\Sigma ,g)$ with ${\int }_{\Sigma }ud{v}_g=0$ and $\Vert {\nabla }_{g}u{\Vert }_2\le 1$, where $W^{1,2}(\Sigma ,g)$ denotes the usual Sobolev space and $\Vert \cdot {\Vert }_2=({\int }_{\Sigma }|\cdot {|}^{2}d{v}_g{)}^{1/2}$ denotes the $L^2(\Sigma ,g)$-norm. This complements our earlier result in [39].

21 pages