Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions
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<+∞, $$\,\underset{{\left(\underset{0}{\overset{1}{\int }}w\left(r\right){\left\vert {u}^{\prime }\le...
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De Gruyter
2025-01-01
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| Series: | Advanced Nonlinear Studies |
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| Online Access: | https://doi.org/10.1515/ans-2023-0161 |
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| author | Xue Jianwei Zhang Caifeng Zhu Maochun |
| author_facet | Xue Jianwei Zhang Caifeng Zhu Maochun |
| author_sort | Xue Jianwei |
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| description | The purpose of this paper is two-fold. First, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions: sup∫01w(r)u′(r)β+2dλα1/(β+2)≤1∫01eμα,θ,γuβ+2β+11−γdλθ<+∞,
$$\,\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. |
| format | Article |
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| language | English |
| publishDate | 2025-01-01 |
| publisher | De Gruyter |
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| series | Advanced Nonlinear Studies |
| spelling | doaj-art-2d438a4d9c8a41238da1c73fae521ef02025-08-20T02:17:46ZengDe GruyterAdvanced Nonlinear Studies2169-03752025-01-0125115217010.1515/ans-2023-0161Trudinger–Moser type inequalities with logarithmic weights in fractional dimensionsXue Jianwei0Zhang Caifeng1Zhu Maochun2School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, P.R. ChinaDepartment of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology of Beijing, Beijing, 100083, P.R. ChinaSchool of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, P.R. ChinaThe 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.https://doi.org/10.1515/ans-2023-0161trudinger–moser inequalitiesfractional dimensionsextremalslogarithmic weight46e3526d1042b37 |
| spellingShingle | Xue Jianwei Zhang Caifeng Zhu Maochun Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions Advanced Nonlinear Studies trudinger–moser inequalities fractional dimensions extremals logarithmic weight 46e35 26d10 42b37 |
| title | Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions |
| title_full | Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions |
| title_fullStr | Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions |
| title_full_unstemmed | Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions |
| title_short | Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions |
| title_sort | trudinger moser type inequalities with logarithmic weights in fractional dimensions |
| topic | trudinger–moser inequalities fractional dimensions extremals logarithmic weight 46e35 26d10 42b37 |
| url | https://doi.org/10.1515/ans-2023-0161 |
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