A sharp Trudinger-Moser type inequality for unbounded domains in ℝⁿ

2006/09/22 by Li, Yuxiang, Ruf, Bernhard · 3 citations
#35J50 #46E35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.math/0609648

Abstract

The Trudinger-Moser inequality states that for functions u ∈ H₀^1,n(Ω) (Ω⊂ \mathbb Rⁿ a bounded domain) with ∫_Ω|∇ u|ⁿdx ≤ 1 one has ∫_Ω(e^α_n|u|^\frac nn-1-1)dx ≤ c |Ω|, with c independent of u. Recently, the second author has shown that for n = 2 the bound c |Ω| may be replaced by a uniform constant d independent of Ω if the Dirichlet norm is replaced by the Sobolev norm, i.e. requiring ∫_Ω(|∇ u|ⁿ + |u|ⁿ)dx ≤ 1. We extend here this result to arbitrary dimensions n > 2. Also, we prove that for Ω= \mathbb Rⁿ the supremum of ∫_{\mathbb Rⁿ} (e^α_n|u|^\frac nn-1-1)dx over all such functions is attained. The proof is based on a blow-up procedure.

Cited by

• The singular anisotropic Adams' type inequality in ℝⁿ
• On the thresholds of the critical singular anisotropic Trudinger–Moser inequality
• Existence of extremal functions and Wulff symmetry for anisotropic Trudinger-Moser inequalities

Related

• Improved Moser--Trudinger inequality for functions with mean value zero in \mathbb Rⁿ and its extremal functions
• A sharp Trudinger-Moser type inequality involving Lⁿ norm in the entire space ℝⁿ
• Singular Trudinger--Moser inequality involving L^p norm in bounded domain
• On the bounds of sharp Trudinger-Moser inequalities
• Remarks on the Extremal Functions for the Moser-Trudinger Inequalities