Anisotropic Moser-Trudinger inequality involving Lⁿ norm in the entire space ℝⁿ

2020/05/12 by Xie, Rulong
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.06513

Abstract

Let F: ℝⁿ→ [0,+∞) be a convex function of class C²( ℝⁿ\backslash\0\) which is even and positively homogeneous of degree 1, and its polar F⁰ represents a Finsler metric on ℝⁿ. The anisotropic Sobolev norm in W^1,n(ℝⁿ) is defined by ||u||_F=(∫_ℝⁿFⁿ(∇ u)+|u|ⁿ)^(1)/(n). In this paper, the following sharp anisotropic Moser-Trudinger inequality involving Lⁿ norm \[ \undersetu∈ W^1,n( ℝⁿ),\Vert u\Vert _F≤ 1sup∫ _{ℝ ⁿ}Φ( λ_n\vert u\vert ^(n)/(n-1)( 1+α\Vert u\Vert _nⁿ) ^(1)/(n-1)) dx0 is sufficiently small. The proof of main results in this paper is based on the method of blow-up analysis.

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