Anisotropic singular Trudinger-Moser inequalities on the whole Euclidean space

Abstract

Let $ F: \mathbb{R}^n\rightarrow [0, \, \infty) $ be a convex function of class $ C^2(\mathbb{R}^n\backslash\{0\}) $, which is even and positively homogeneous of degree $ 1 $. In this paper, we prove that$ \sup\limits_{u\in W^{1, n}(\mathbb{R}^n), \, \displaystyle{\int}_{\mathbb{R}^n}(F^n(\nabla u)+|u|^n)dx\leq1}\displaystyle{\int}_{\mathbb{R}^n}\frac{\Phi(\lambda_{n}(1-\frac{\beta}{n})(1+\alpha\|u\|^{n}_n)^{\frac{1}{n-1}}|u|^{\frac{n}{n-1}})}{F^o(x)^\beta}dx $is finite for $ 0\leq\alpha<1 $, and the supremum is infinity for $ \alpha\geq1 $, where $ F^o(x) $ is the polar function of $ F $, $ \Phi(t) = e^t-\sum_{j = 0}^{n-2}\frac{t^j}{j!} $, $ \beta\in[0, n) $, $ \lambda_n = n^{\frac{n}{n-1}}\kappa_n^{\frac{1}{n-1}} $ and $ \kappa_n $ is the volume of the unit Wulff ball. Moreover, by using the method of blow-up analysis, we also obtain the existence of extremal functions for the supremum when $ 0\leq\alpha<1 $.