Existence of extremal functions and Wulff symmetry for anisotropic Trudinger-Moser inequalities

TL;DR

This paper studies the existence and symmetry of extremal functions for anisotropic Trudinger-Moser inequalities using convex symmetrization and continuity methods.

Contribution

It provides new results on the existence and symmetry of extremal functions for various anisotropic Trudinger-Moser inequalities.

Findings

Existence of extremal functions established.

Symmetry properties of extremal functions demonstrated.

Relations between subcritical and critical supremums analyzed.

Abstract

In this paper, we investigate the extremal functions for anisotropic Trudinger-Moser inequalities. Our method uses convex symmetrization, the continuity of the supremum function, together with the relation between the supremums of the subcritical and the critical anisotropic Trudinger-Moser inequality, we give some results of existence and symmetry about the extremal functions for several different types of anisotropic Trudinger-Moser inequalities.