Weighted anisotropic isoperimetric inequalities and existence of extremals for singular anisotropic Trudinger-Moser inequalities

Abstract

In this paper, we establish a class of isoperimetric inequalities on $R^n$ with respect to weights which are negative powers of the distance to the origin associated with the Finsler metric. (See Theorem 1.1.) Based on these weighted anisotropic isoperimetric inequalities, we can classify a class of singular Liouville's equation associated with the n-Finsler-Laplacian (1.10) and construct a blow-up sequence to show the existence of extremals for the singular Trudinger-Moser inequality involving the anisotropic Dirichlet norm:$\underset{u\in W_0^{1,n}\left(\Omega \right),{\int }_{\Omega }F{\left(\nabla u\right)}^{n}dx\le 1}{\sup }\underset{\Omega }{\int }{e}^{{\lambda }_{n,\beta }|u{|}^{\frac{n}{n-1}}}{F}^{\circ }{\left(x\right)}^{-\beta }dx<\infty$ for any $\beta \in (0,n)$, where $\Omega \subset R^n$ is a smooth and bounded domain containing the origin, and ${\lambda }_{n,\beta }:=\left(\frac{n-\beta }{n}\right)n^{\frac{n}{n-1}}{\kappa }_n^{\frac{1}{n-1}}$. Here F is a convex function, which is even and positively homogeneous of degree 1, and its polar $F^{\circ }$ represents a Finsler metric on $R^n,{\kappa }_n$ is the Lebesgue measure of the unit Wulff ball.

The presence of the weight in Theorem 1.1 adds significant difficulties because of the lack of the appropriate symmetrization principle with the weight function. To this end, we perform a novel quasi-conformal type of map associated with the Finsler metric to deal with this weight. Theorem 1.1 is crucial in classifying the solutions to a class of singular Liouville's equation associated with the n-Finsler-Laplacian (1.10). (See Theorem 4.1.) This classification plays an important role in establishing the existence of extremal functions to the singular anisotropic Trudinger-Moser inequality. (See Theorem 1.2.)