A sharp form of Moser–Trudinger inequality in high dimension

Abstract

Let Ω be a bounded smooth domain in R n (n 3). This paper deals with a sharp form of MoserTrudinger inequality. Let λ 1 (Ω) = inf u∈H 1,n 0 (Ω), u ≡0

Multibubble analysis on N-Laplace equation in $${\mathbb{R}^N}$$

“…We should mention that one bubble analysis for solutions to N -Laplace equations has been considered by Li [9] and the second author [19,20]. Similar results has been obtained by Martinazzi [10] for powers of the Laplacian.…”

Extremal functions for a singular Hardy-Moser-Trudinger inequality

“…whereas for any α ≥ λ 1 (Ω), the above supremum is infinity. The analogs of (2) were obtained on a compact Riemannian surface [23] and on a high dimeniosnal Euclidean domain [24]. Clearly, the inequality (2) is stronger than the inequality (1).…”

Anisotropic Moser-Trudinger inequality involving L norm

“…In other words, there exists u α ∈ H ∩ C 1 (Ω) such that J α λn (u α ) = sup u∈H J α λn (u). When F (x) = |x|, Adimurthi and Druet [1], Y.Y Yang [38] have proved the above theorem. But the F (x) = |x|, it is more different, need much more delicate work.…”

“…where G and ψ are functions given in (38), R = − log ǫ, η ∈ C 1 0 (W 2Rǫ (x 0 )) satisfying that η = 1 on W Rǫ (x 0 ) and |∇η| ≤ 2 Rǫ , b and C are constants depending only on ǫ to be determined later. Clearly W 2Rǫ (x 0 ) ⊂ Ω provided that ǫ is sufficiently small.…”