Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions - Xue Jianwei; Zhang Caifeng; Zhu Maochun (Advanced Nonlinear Studies, 2025)
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<+∞, $$\,\underset{{\left(\underset{0}{\overset{1}{\int }}w\left(r\right){\left\vert {u}^{\prime }\left(r\right)\right\vert }^{\beta +2}\mathrm{d}{\lambda }_{\alpha }\right)}^{1/\left(\beta +2\right)}\le 1}{\mathrm{sup}}\underset{0}{\overset{1}{\int }}{\text{e}}^{{\mu }_{\alpha ,\theta ,\gamma }{\left\vert u\right\vert }^{\frac{\beta +2}{\left(\beta +1\right)\left(1-\gamma \right)}}}\mathrm{d}{\lambda }_{\theta }{< }+\infty ,$$ where 0 ≤ γ < 1, α = β + 1, μα,θ,γ≔θ+1ωα1/α1−γ11−γ ${\mu }_{\alpha ,\theta ,\gamma }{:=}\left(\theta +1\right){\left[{\omega }_{\alpha }^{1/\alpha }\left(1-\gamma \right)\right]}^{\frac{1}{1-\gamma }}$ , w(r)=w1(r)=log1rγβ+1 $w\left(r\right)={w}_{1}\left(r\right)={\left(\mathrm{log}\frac{1}{r}\right)}^{\gamma \left(\beta +1\right)}$ or w(r)=w2(r)=logerγβ+1 $w\left(r\right)={w}_{2}\left(r\right)={\left(\mathrm{log}\frac{e}{r}\right)}^{\gamma \left(\beta +1\right)}$ and λ θ(E) = ω θ ∫ E r θdr for all E⊂R $E\subset \mathbb{R}$ . The case γ > 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.