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

Let F : R-n -> [0,+ infinity) be a convex function of class C-2(R-n\{0}) which is even and positively homogeneous of degree 1, and its polar F-0 represents a Finsler metric on R-n. The anisotropic Sobolev norm in W-1,W-n(R-n) is defined by parallel to u parallel to F = (integral(Rn) (F-n(del u) + vertical bar u vertical bar(n))dx)(1/n). In this paper, the following sharp anisotropic Moser-Trudinger inequality involving L-n norm sup(u is an element of W1,n(Rn)),(parallel to u parallel to F <= 1) integral(Rn) Phi(lambda(n)vertical bar u vertical bar n/n-1 (1 + alpha parallel to u parallel to(n)(n))(1/n-1))dx < +infinity in the entire space R-n for any 0 < alpha < 1 is established, where Phi(t) = e(t) - Sigma(n-2)(j=0) t(j)/j(1), lambda(n) = n n/n-1 kappa(n) 1/n-1 and kappa(n) is the volume of the unit Wulff ball in R-n. It is also shown that the above supremum is infinity for all alpha >= 1. Moreover, we prove the supremum is attained, that is, there exists a maximizer for the above supremum when alpha > 0 is sufficiently small.