Anisotropic Moser-Trudinger inequality involving L norm

An improved Trudinger-Moser inequality involving N-Finsler-Laplacian and L^p norm

“…for the special case p = N in (1.6), we refer reader to [57]. Next, let's show that we have got better result.…”

“…This operator Q N was studied in some literatures, see [5,6,18] and the references therein. In [56], they obtained the existence of extremal functions for the sharp geometric inequality (1.4).…”

“…1 is uniformly bounded in L q (Ω) for some q > 1. Then by Lemma 2.2 in [56], u ǫ is uniformly bounded in Ω, which contradicts c ǫ → +∞ as ǫ → 0. Hence u 0 ≡ 0.…”

“…We will establish the Lions type concentration-compactness principle [30] for Trudinger-Moser Inequalities under anisotropic Dirichlet norm and L p norm, which is the extention of Theorem 1.1 in [9] and Lemma 2.3 in [56].…”

“…Finally, replacing the right hand term in equation (4.11), we use the similar discussion as Lemma 4.7 in[57], the asymptotic representation of Green function can immediateiy derived. This complete the proof of Lemma 4.5.…”