Sharp Singular Trudinger–Moser Inequalities Under Different Norms

Abstract

The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in ℝN{\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces DN,q⁢(ℝN){D^{N,q}(\mathbb{R}^{N})}, q≥1{q\geq 1}, the completion of C0∞⁢(ℝN){C_{0}^{\infty}(\mathbb{R}^{N})} under the norm ∥∇⁡u∥N+∥u∥q{\|\nabla u\|_{N}+\|u\|_{q}}. The case q=N{q=N} (i.e., DN,q⁢(ℝN)=W1,N⁢(ℝN){D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})}) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type ∥∇⁡u∥N≤1{\|\nabla u\|_{N}\leq 1} and full-norm type ∥∇⁡u∥Na+∥u∥qb≤1{\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1}, a>0{a>0}, b>0{b>0}. We will show that the Trudinger–Moser-type inequalities hold if and only if b≤N{b\leq N}. Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when b>N{b>N}.

Keywords

• trudinger–moser inequality
• maximizers
• sharp constants
• exact growth
• 26d10
• 42b35
• 46e35