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Journal articles on the topic 'Moser-Trudinger'

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Published: 14 March 2023

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1

Song, Xiaoping, Dongliang Li, and Maochun Zhu. "Critical and Subcritical Anisotropic Trudinger–Moser Inequalities on the Entire Euclidean Spaces." Mathematical Problems in Engineering 2021 (September 30, 2021): 1–13. http://dx.doi.org/10.1155/2021/8992411.

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We investigate the subcritical anisotropic Trudinger–Moser inequality in the entire space ℝ N , obtain the asymptotic behavior of the supremum for the subcritical anisotropic Trudinger–Moser inequalities on the entire Euclidean spaces, and provide a precise relationship between the supremums for the critical and subcritical anisotropic Trudinger–Moser inequalities. Furthermore, we can prove critical anisotropic Trudinger–Moser inequalities under the nonhomogenous norm restriction and obtain a similar relationship with the supremums of subcritical anisotropic Trudinger–Moser inequalities.

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2

Kim, Meelae. "Linearised Moser-Trudinger inequality." Bulletin of the Australian Mathematical Society 62, no. 3 (December 2000): 445–57. http://dx.doi.org/10.1017/s0004972700018967.

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As a limiting case of the Sobolev imbedding theorem, the Moser-Trudinger inequality was obtained for functions in with resulting exponential class integrability. Here we prove this inequality again and at the same time get sharper information for the bound. We also generalise the Linearised Moser inequality to higher dimensions, which was first introduced by Beckner for functions on the unit disc. Both of our results are obtained by using the method of Carleson and Chang. The last section introduces an analogue of each inequality for the Laplacian instead of the gradient under some restricted conditions.

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3

Cianchi, Andrea. "Moser–Trudinger trace inequalities." Advances in Mathematics 217, no. 5 (March 2008): 2005–44. http://dx.doi.org/10.1016/j.aim.2007.09.007.

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4

MANCINI, G., and K. SANDEEP. "MOSER–TRUDINGER INEQUALITY ON CONFORMAL DISCS." Communications in Contemporary Mathematics 12, no. 06 (December 2010): 1055–68. http://dx.doi.org/10.1142/s0219199710004111.

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We prove that a sharp Moser–Trudinger inequality holds true on a conformal disc if and only if the metric is bounded from above by the Poincaré metric. We also derive necessary and sufficient conditions for the validity of a sharp Moser–Trudinger inequality on a simply connected domain in ℝ2.

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5

Hyder, Ali. "Moser functions and fractional Moser–Trudinger type inequalities." Nonlinear Analysis 146 (November 2016): 185–210. http://dx.doi.org/10.1016/j.na.2016.08.024.

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6

Wang, Guofang, and Dong Ye. "A Hardy–Moser–Trudinger inequality." Advances in Mathematics 230, no. 1 (May 2012): 294–320. http://dx.doi.org/10.1016/j.aim.2011.12.001.

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7

del Pino, Manuel, Monica Musso, and Bernhard Ruf. "Beyond the Trudinger-Moser supremum." Calculus of Variations and Partial Differential Equations 44, no. 3-4 (August 25, 2011): 543–76. http://dx.doi.org/10.1007/s00526-011-0444-5.

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8

Dolbeault, Jean, Maria J. Esteban, and Gaspard Jankowiak. "The Moser-Trudinger-Onofri inequality." Chinese Annals of Mathematics, Series B 36, no. 5 (August 7, 2015): 777–802. http://dx.doi.org/10.1007/s11401-015-0976-7.

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9

Santaria Leuyacc, Yony Raúl. "Nonlinear Elliptic Equations with Maximal Growth Range." Pesquimat 20, no. 1 (September 4, 2017): 1. http://dx.doi.org/10.15381/pes.v20i1.13753.

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En este trabajo nos interesa estudiar la existencia de soluciones débiles no triviales para una clase de ecuaciones elípticas no lineales definidas en un dominio limitado en dimensión dos, donde las no linealidades poseen un rango de crecimiento exponencial máximo motivado por las desigualdades de Trudinger-Moser en espacios de Lorentz-Sobolev. Para estudiar la solubilidad se utiliza un enfoque variacional. Más específicamente, usamos el teorema del paso de montaña junto con desigualdades de tipo Trudinger-Moser.

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10

Li, Jungang, Guozhen Lu, and Maochun Zhu. "Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument." Advanced Nonlinear Studies 21, no. 4 (October 10, 2021): 917–37. http://dx.doi.org/10.1515/ans-2021-2147.

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Abstract The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

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11

郭, 明娟. "Moser-Trudinger Inequalities in Hyperbolic Spaces." Advances in Applied Mathematics 10, no. 02 (2021): 453–60. http://dx.doi.org/10.12677/aam.2021.102051.

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12

Fang, Fei, and Chao Ji. "The cone Moser–Trudinger inequalities and applications." Asymptotic Analysis 120, no. 3-4 (October 30, 2020): 273–99. http://dx.doi.org/10.3233/asy-191588.

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In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in x ∈ int ( B ) , u = 0 , on ∂ B , where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0, and the nonlinear term f has the subcritical exponential growth or the critical exponential growth.

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13

Jevnikar, Aleks, and Wen Yang. "A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 2 (December 27, 2018): 325–52. http://dx.doi.org/10.1017/prm.2018.30.

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AbstractWe are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

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14

Yu, Peng-Xiu. "A Weighted Singular Trudinger-Moser Inequality." Journal of Partial Differential Equations 35, no. 3 (June 2022): 208–22. http://dx.doi.org/10.4208/jpde.v35.n3.2.

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15

Wang, Guofang. "Moser-Trudinger inequalities and Liouville systems." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 328, no. 10 (May 1999): 895–900. http://dx.doi.org/10.1016/s0764-4442(99)80293-6.

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16

Calanchi, M., and B. Ruf. "Weighted Trudinger - Moser Inequalities and Applications." Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming and Computer Software" 8, no. 3 (2015): 42–55. http://dx.doi.org/10.14529/mmp150303.

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17

Martinazzi, Luca. "Fractional Adams–Moser–Trudinger type inequalities." Nonlinear Analysis: Theory, Methods & Applications 127 (November 2015): 263–78. http://dx.doi.org/10.1016/j.na.2015.06.034.

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18

Tintarev, Cyril. "Trudinger–Moser inequality with remainder terms." Journal of Functional Analysis 266, no. 1 (January 2014): 55–66. http://dx.doi.org/10.1016/j.jfa.2013.09.009.

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19

Yang, Yunyan, and Xiaobao Zhu. "An improved Hardy–Trudinger–Moser inequality." Annals of Global Analysis and Geometry 49, no. 1 (September 25, 2015): 23–41. http://dx.doi.org/10.1007/s10455-015-9478-9.

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20

Zhang, Caifeng. "Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation." Advanced Nonlinear Studies 19, no. 1 (February 1, 2019): 197–217. http://dx.doi.org/10.1515/ans-2018-2026.

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Abstract In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in {W^{s,p}(\mathbb{R}^{N})} . Define \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0. There holds \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty, where {s\in(0,1)} , {sp=N} , {\alpha\in[0,\alpha_{*})} and \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where {V(x)} has a positive lower bound, {f(x,t)} behaves like {e^{\alpha\lvert t\rvert^{N/(N-s)}}} , {h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and {\varepsilon>0} . Moreover, we also derive a weak solution with negative energy.

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21

Lam, Nguyen, Guozhen Lu, and Lu Zhang. "Sharp Singular Trudinger–Moser Inequalities Under Different Norms." Advanced Nonlinear Studies 19, no. 2 (May 1, 2019): 239–61. http://dx.doi.org/10.1515/ans-2019-2042.

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AbstractThe main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in {\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces {D^{N,q}(\mathbb{R}^{N})}, {q\geq 1}, the completion of {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm {\|\nabla u\|_{N}+\|u\|_{q}}. The case {q=N} (i.e., {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 {\|\nabla u\|_{N}\leq 1} and full-norm type {\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1}, {a>0}, {b>0}. We will show that the Trudinger–Moser-type inequalities hold if and only if {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}.

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22

Sandeep, Kunnath. "Moser–Trudinger–Adams inequalities and related developments." Bulletin of Mathematical Sciences 10, no. 02 (August 2020): 2030001. http://dx.doi.org/10.1142/s1664360720300017.

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In this paper, we review some of the developments on the Moser–Trudinger and Adams inequalities in both Euclidean space and on Riemannian manifolds. We will also describe some of the closely related problems.

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23

Lam, Nguyen. "Equivalence of sharp Trudinger-Moser-Adams Inequalities." Communications on Pure & Applied Analysis 16, no. 3 (2017): 973–98. http://dx.doi.org/10.3934/cpaa.2017047.

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24

Zhu, Xiaobao. "Remarks on singular trudinger-moser type inequalities." Communications on Pure & Applied Analysis 19, no. 1 (2020): 103–12. http://dx.doi.org/10.3934/cpaa.2020006.

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25

Yu, Luo Qianjin and Fang. "A Remark on Hardy-Trudinger-Moser Inequality." Journal of Partial Differential Equations 31, no. 4 (June 2018): 353–73. http://dx.doi.org/10.4208/jpde.v31.n4.6.

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26

Iula, Stefano, and Gabriele Mancini. "Extremal functions for singular Moser–Trudinger embeddings." Nonlinear Analysis 156 (June 2017): 215–48. http://dx.doi.org/10.1016/j.na.2017.02.029.

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27

Cianchi, Andrea, Erwin Lutwak, Deane Yang, and Gaoyong Zhang. "Affine Moser–Trudinger and Morrey–Sobolev inequalities." Calculus of Variations and Partial Differential Equations 36, no. 3 (April 10, 2009): 419–36. http://dx.doi.org/10.1007/s00526-009-0235-4.

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28

Tian, Gang, and Xiaohua Zhu. "A nonlinear inequality of Moser-Trudinger type." Calculus of Variations and Partial Differential Equations 10, no. 4 (June 1, 2000): 349–54. http://dx.doi.org/10.1007/s005260010349.

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29

Fang, Hao. "On a Multi-particle Moser-Trudinger Inequality." Communications in Analysis and Geometry 12, no. 5 (2004): 1155–72. http://dx.doi.org/10.4310/cag.2004.v12.n5.a8.

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30

Yang, Yunyan, and Xiaobao Zhu. "Trudinger-Moser Embedding on the Hyperbolic Space." Abstract and Applied Analysis 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/908216.

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Let(ℍn,g)be the hyperbolic space of dimensionn. By our previous work (Theorem 2.3 of (Yang (2012))), for any0<α<αn, there exists a constantτ>0depending only onnandαsuch thatsupu∈W1,n(ℍn),∥u∥1,τ≤1∫ℍn(eαun/(n-1)-∑k=0n-2αk|u|nk/(n-1)/k!)dvg<∞,whereαn=nωn-11/(n-1),ωn-1is the measure of the unit sphere inℝn, andu1,τ=∇guLn(ℍn)+τuLn(ℍn). In this note we shall improve the above mentioned inequality. Particularly, we show that, for any0<α<αnand anyτ>0, the above mentioned inequality holds with the definition ofu1,τreplaced by(∫ℍn‍(|∇gu|n+τ|u|n)dvg)1/n. We solve this problem by gluing local uniform estimates.

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31

Adimurthi and Przemysław Górka. "Global Trudinger-Moser inequality on metric spaces." Mathematical Inequalities & Applications, no. 3 (2016): 1131–39. http://dx.doi.org/10.7153/mia-19-83.

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32

Černý, Robert. "Moser-Trudinger inequality in grand Lebesgue space." Rendiconti Lincei - Matematica e Applicazioni 26, no. 2 (2015): 177–88. http://dx.doi.org/10.4171/rlm/701.

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33

Zhou, Changliang. "Anisotropic Moser-Trudinger inequality involving L norm." Journal of Differential Equations 268, no. 12 (June 2020): 7251–85. http://dx.doi.org/10.1016/j.jde.2019.11.066.

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34

Battaglia, Luca. "Moser–Trudinger inequalities for singular Liouville systems." Mathematische Zeitschrift 282, no. 3-4 (November 23, 2015): 1169–90. http://dx.doi.org/10.1007/s00209-015-1584-7.

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35

Nguyen, Van Hoang. "Improved critical Hardy inequality and Leray-Trudinger type inequalities in Carnot groups." Annales Fennici Mathematici 47, no. 1 (December 2, 2021): 121–38. http://dx.doi.org/10.54330/afm.112567.

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In this paper, we prove an improvement of the critical Hardy inequality in Carnot groups. We show that this improvement is sharp and can not be improved. We apply this improved critical Hardy inequality together with the Moser-Trudinger inequality due to Balogh, Manfredi and Tyson (2003) to establish the Leray-Trudinger type inequalities which extend the inequalities of Psaradakis and Spector (2015) and Mallick and Tintarev (2018) to the setting of Carnot groups.

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36

Abid, Imed, Sami Baraket, and Rached Jaidane. "On a weighted elliptic equation of N-Kirchhoff type with double exponential growth." Demonstratio Mathematica 55, no. 1 (January 1, 2022): 634–57. http://dx.doi.org/10.1515/dema-2022-0156.

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Abstract In this work, we study the weighted Kirchhoff problem − g ∫ B σ ( x ) ∣ ∇ u ∣ N d x div ( σ ( x ) ∣ ∇ u ∣ N − 2 ∇ u ) = f ( x , u ) in B , u > 0 in B , u = 0 on ∂ B , \left\{\begin{array}{ll}-g\left(\mathop{\displaystyle \int }\limits_{B}\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N}{\rm{d}}x\right){\rm{div}}\left(\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N-2}\nabla u)=f\left(x,u)& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial B,\end{array}\right. where B B is the unit ball of R N {{\mathbb{R}}}^{N} , σ ( x ) = log e ∣ x ∣ N − 1 \sigma \left(x)={\left(\log \left(\frac{e}{| x| }\right)\right)}^{N-1} , the singular logarithm weight in the Trudinger-Moser embedding, and g g is a continuous positive function on R + {{\mathbb{R}}}^{+} . The nonlinearity is critical or subcritical growth in view of Trudinger-Moser inequalities. We first obtain the existence of a solution in the subcritical exponential growth case with positive energy by using minimax techniques combined with the Trudinger-Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth, and we stress its importance to check the compactness level.

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37

Zhang, Guoqing, and Jing Sun. "Ground-State Solutions for a Class ofN-Laplacian Equation with Critical Growth." Abstract and Applied Analysis 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/831468.

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We investigate the existence of ground-state solutions for a class ofN-Laplacian equation with critical growth inℝN. Our proof is based on a suitable Trudinger-Moser inequality, Pohozaev-Pucci-Serrin identity manifold, and mountain pass lemma.

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38

Nguyen, Van Hoang. "Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in W1,N(ℝN)." Communications in Contemporary Mathematics 21, no. 04 (May 31, 2019): 1850023. http://dx.doi.org/10.1142/s0219199718500232.

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We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] both in the subcritical case [Formula: see text] and critical case [Formula: see text] with [Formula: see text] and [Formula: see text] denotes the surface area of the unit sphere in [Formula: see text]. We will show that MT[Formula: see text] is attained in the subcritical case if [Formula: see text] or [Formula: see text] and [Formula: see text] with [Formula: see text] being the best constant in a Gagliardo–Nirenberg inequality in [Formula: see text]. We also show that MT[Formula: see text] is not attained for [Formula: see text] small which is different from the context of bounded domains. In the critical case, we prove that MT[Formula: see text] is attained for [Formula: see text] small enough. To prove our results, we first establish a lower bound for MT[Formula: see text] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[Formula: see text] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[Formula: see text]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [Formula: see text], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.].

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39

YANG, YUNYAN. "EXTREMAL FUNCTIONS FOR MOSER–TRUDINGER INEQUALITIES ON 2-DIMENSIONAL COMPACT RIEMANNIAN MANIFOLDS WITH BOUNDARY." International Journal of Mathematics 17, no. 03 (March 2006): 313–30. http://dx.doi.org/10.1142/s0129167x06003473.

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Let (M,g) be a 2-dimensional compact Riemannian manifold with boundary. In this paper, we use the method of blowing up analysis to prove the existence of the extremal functions for some Moser–Trudinger inequalities on (M,g).

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40

do Ó, João Marcos, and José Francisco de Oliveira. "Concentration-compactness and extremal problems for a weighted Trudinger–Moser inequality." Communications in Contemporary Mathematics 19, no. 01 (November 24, 2016): 1650003. http://dx.doi.org/10.1142/s0219199716500036.

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We discuss critical and subcritical general maximizing problems associated with a Trudinger–Moser type inequality for a class of weighted Sobolev spaces. The key of our approach relies on a concentration-compactness type result and sharp estimates for the concentration level of a Moser-type functional. In particular, we give an explicit concentrating sequence that is maximizing for the maximal concentration level. As an application, we prove the existence of a nontrivial solution for a family of quasilinear elliptic equations with critical growth.

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41

Pruss, Alexander R. "Nonexistence of Maxima for Perturbations of Some Inequalities with Critical Growth." Canadian Mathematical Bulletin 39, no. 2 (June 1, 1996): 227–37. http://dx.doi.org/10.4153/cmb-1996-029-1.

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AbstractWe study the question of nonexistence of extremal functions for perturbations of some sharp inequalities such as those of Moser-Trudinger (1971) and Chang- Marshall (1985). We shall show that for each critically sharp (in a sense that will be precisely defined) inequality of the formwhere is a collection of measurable functions on a finite measure space (I, μ) and O a nonnegative continuous function on [0, ∞), we have a continuous Ψ on [0, ∞) with 0 ≤ Ψ ≤ Φ, but withnot being attained even if the supremum in (1) is attained. We then apply our results to the Moser-Trudinger and Chang-Marshall inequalities. Our result is to be contrasted with the fact shown by Matheson and Pruss (1994) that if Ψ(t) = o(Φ(t) as t —> ∞ then the supremum in (2) is attained. In the present paper, we also give a converse to that fact.

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42

Mancini, Gabriele, and Luca Martinazzi. "Extremals for Fractional Moser–Trudinger Inequalities in Dimension 1 via Harmonic Extensions and Commutator Estimates." Advanced Nonlinear Studies 20, no. 3 (August 1, 2020): 599–632. http://dx.doi.org/10.1515/ans-2020-2089.

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AbstractWe prove the existence of extremals for fractional Moser–Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler–Lagrange equation, which requires new sharp estimates obtained via commutator techniques.

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43

Xiaobao, Zhu. "A Singular Trudinger-Moser Inequality in Hyperbolic Space." Journal of Partial Differential Equations 28, no. 1 (June 2015): 39–46. http://dx.doi.org/10.4208/jpde.v28.n1.5.

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44

Cingolani, Silvia, and Tobias Weth. "Trudinger–Moser‐type inequality with logarithmic convolution potentials." Journal of the London Mathematical Society 105, no. 3 (February 15, 2022): 1897–935. http://dx.doi.org/10.1112/jlms.12549.

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45

D. H. Phong, Jian Song, Jacob Sturm, and Ben Weinkove. "The Moser-Trudinger inequality on Kähler-Einstein manifolds." American Journal of Mathematics 130, no. 4 (2008): 1067–85. http://dx.doi.org/10.1353/ajm.0.0013.

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46

Csató, Gyula, and Prosenjit Roy. "Singular Moser–Trudinger inequality on simply connected domains." Communications in Partial Differential Equations 41, no. 5 (November 30, 2015): 838–47. http://dx.doi.org/10.1080/03605302.2015.1123276.

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Su, Dan, and Qiaohua Yang. "Trudinger–Moser inequalities on harmonicANgroups under Lorentz norms." Nonlinear Analysis 188 (November 2019): 439–54. http://dx.doi.org/10.1016/j.na.2019.06.010.

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Calanchi, Marta, and Bernhard Ruf. "On Trudinger–Moser type inequalities with logarithmic weights." Journal of Differential Equations 258, no. 6 (March 2015): 1967–89. http://dx.doi.org/10.1016/j.jde.2014.11.019.

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Adimurthi and K. Sandeep. "A singular Moser-Trudinger embedding and its applications." Nonlinear Differential Equations and Applications NoDEA 13, no. 5-6 (July 2007): 585–603. http://dx.doi.org/10.1007/s00030-006-4025-9.

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Yang, Yunyan. "Trudinger–Moser inequalities on complete noncompact Riemannian manifolds." Journal of Functional Analysis 263, no. 7 (October 2012): 1894–938. http://dx.doi.org/10.1016/j.jfa.2012.06.019.

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