Compactness of Extremals for Singular Anisotropic Trudinger-Moser functionals on bounded domain

2025/12/08 by Shan, Weiwei, Yang, Minbo, Zhou, Jiazheng
#46E38 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.07118

Abstract

In this paper, we investigate the compactness of extremal functions for a critical singular anisotropic Trudinger-Moser inequality established by Lu-Shen-Xue-Zhu\citeref1. We prove by means of blow-up analysis that the extremals u_β converge in W₀^1,n(Ω)∩ C¹(Ω) to some function u₀ which achieves the supremum sup_u∈ W₀^1,n(Ω),\Vert u\Vert_F(Ω)≤1∫_Ωe^τ_n\vert u\vert^(n)/(n-1)dx,as β→ 0, where τ_n=n^(n)/(n-1)κ_n^(1)/(n-1), κ_n denotes the volume of the unit Wulff ball in ℝⁿ and \Vert u\Vert_F(Ω) is the anisotropic norm of u.

Citations

• Sharp anisotropic singular Trudinger-Moser inequalities in the entire space
• Concentration-compactness principle of singular Trudinger-Moser inequality involving N-Finsler-Laplacian operator
• Concentration-compactness principle of singular Trudinger–Moser inequality involving N-Finsler–Laplacian operator
• A Trudinger-Moser inequality for conical metric in the unit ball
• Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation
• Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space
• Boundary regularity for solutions of degenerate elliptic equations

Related

• On the thresholds of the critical singular anisotropic Trudinger–Moser inequality
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• Extremal functions for the sharp Moser--Trudinger type inequalities in\n whole space mathbb R^N
• Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space
• A simplification of the proof of the existence of the extremal function for the Moser-Trudinger inequality