Anisotropic Moser–Trudinger Inequality Involving Ln Norm in the Entire Space ℝn.

Let F: ℝn → [0, + ∞) be a convex function of class C2(ℝn{0}) which is even and positively homogeneous of degree 1, and its polar F0 represents a Finsler metric on ℝn. The anisotropic Sobolev norm in W1,n(ℝn) is defined by ‖ u ‖ F = ( ∫ ℝ n (F n (∇ u) + | u | n) d x ) 1 / n. In this paper, the following sharp anisotropic Moser–Trudinger inequality involving Ln norm sup u ∈ W 1 , n (ℝ n) , ‖ u ‖ F ≤ 1 ∫ ℝ n Φ (λ n | u | n n − 1 (1 + α ‖ u ‖ n n) 1 n − 1 ) d x < + ∞ in the entire space ℝn for any 0 < α < 1 is established, where Φ (t) = e t − ∑ j = 0 n − 2 t j j ! , λ n = n n n − 1 κ n 1 n − 1 and κn is the volume of the unit Wulff ball in ℝn. It is also shown that the above supremum is infinity for all α ≥ 1. Moreover, we prove the supremum is attained, that is, there exists a maximizer for the above supremum when α > 0 is sufficiently small. [ABSTRACT FROM AUTHOR]