Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions

Abstract

The purpose of this paper is two-fold. First, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions: sup∫01w(r)u′(r)β+2dλα1/(β+2)≤1∫01eμα,θ,γuβ+2β+11−γdλθ 1 and γ = 1 are also be considered in this part to improve our paper. Indeed, we have a continuous embedding X(w 2) ↪ L ∞(0, 1) for γ > 1 and a critical growth of double exponential type for γ = 1. Second, we apply the Lions type Concentration-Compactness principle for Trudinger–Moser inequalities and the precise estimate of normalized concentration limit for normalized concentrating sequence at origin to establish the existence of extremals for Trudinger–Moser inequalities when w(r)=w1(r)=log1rγβ+1 $w\left(r\right)={w}_{1}\left(r\right)={\left(\mathrm{log}\frac{1}{r}\right)}^{\gamma \left(\beta +1\right)}$ and γ > 0 is sufficiently small.

Keywords

• trudinger–moser inequalities
• fractional dimensions
• extremals
• logarithmic weight
• 46e35
• 26d10
• 42b37