A Sharp Moser–Trudinger Type Inequality Involving Adimurthi–Druet Term in Two Dimensional Hyperbolic Space

Let B g ( 1 ) ⊂ H 2 be the unit geodesic ball and W 0 1 , 2 be the standard Sobolev space. Let λ 1 ( B g ( 1 ) ) be the first Dirichlet eigenvalue in B g ( 1 ) associated to the Laplace–Beltrami operator - Δ g on H 2 . In this paper, we establish a sharp Moser–Trudinger inequality involving the Adimurthi–Duret term by a delicate blow-up analysis. Specifically, we prove that sup u ∈ W 0 1 , 2 ( B g ( 1 ) ) , ‖ ∇ g u ‖ 2 = 1 ∫ B g ( 1 ) e 4 π ( 1 + α ‖ u ‖ 2 2 ) u 2 d Vol g < ∞ if and only if α < λ 1 ( B g ( 1 ) ) . We also prove the existence of extremal functions for the above inequality when α is sufficiently small, which, to our knowledge, is the first result on the extremal function of Moser–Trudinger inequality involving Adimurthi–Druet term in Hyperbolic space. Besides, we extend the above results to the entire space H 2 partially, namely, we prove that sup u ∈ W 1 , 2 ( H 2 ) , ‖ ∇ g u ‖ 2 = 1 ∫ H 2 e 4 π ( 1 + α ‖ u ‖ 2 2 ) u 2 - 1 d Vol g is finite if α < 1 / 4 , and is infinite if α > 1 / 4 . If α = 1 / 4 , we derive a different Adimurthi–Druet type result using an effective scaling argument, which is established by Chen et al. in [ 9 ] and [ 10 ].