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 (Σ, g). Precisely, if λ 1 (Σ) is the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition, then there exists a positive real number α * < λ 1 (Σ) such that for all α ∈ (α * , λ 1 (Σ)), the supremumcan not be attained by any u ∈ W 1,2 (Σ, g) with Σ udv g = 0 and ∇ g u 2 ≤ 1, where W 1,2 (Σ, g) denotes the usual Sobolev space and • 2 = ( Σ | • | 2 dv g ) 1/2 denotes the L 2 (Σ, g)-norm. This complements our earlier result in [39].