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

TL;DR

This work addresses the existence and symmetry of extremal functions for anisotropic Trudinger-Moser inequalities in $\, ext{R}^N$. It develops a framework based on convex symmetrization with respect to the Wulff shape, leveraging the continuity of the supremum functional and the link between subcritical and critical inequalities to derive maximizers. The authors prove existence and Wulff symmetry for subcritical ATMSC maximizers, show that maximizers can be taken Wulff symmetric in the singular case, and analyze the critical (subcritical) problem through auxiliary functionals $ ext{ATMSC}$ and $ ext{Λ}_{a,b}$, giving attainment results in several parameter regimes and a non-attainment criterion when $eta=0$ and $rac{q(N-1)}{N}\

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.

Theorems & Definitions (19)

• Theorem A
• Theorem B
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof