Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations

Abstract

In this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger–Moser inequality in ℝn{\mathbb{R}^{n}}. Then we prove a compact embedding theorem, which states that W1,n⁢(ℝn){W^{1,n}(\mathbb{R}^{n})} is compactly embedded into Lp⁢(ℝn,|x|-β⁢d⁢x){L^{p}(\mathbb{R}^{n},|x|^{-\beta}\,dx)} for p≥n{p\geq n} and 0<β<n{0<\beta<n}. As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the following n-Laplace equation with critical nonlinearity:

Keywords

• trudinger–moser inequality
• mountain-pass theorem
• ground state solutions, palais–smale compactness condition
• 35j60
• 35b33
• 46e30