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Existence of an extremal function of critical Sobolev embedding with an α-homogeneous weight
[Existence d’une fonction extrémale d’une injéction de Sobolev critique avec des poids homogènes d’ordre α]
Petr Gurka1, 2, 3  ; Daniel Hauer4, 5 
1 Department of Mathematics, Czech University of Life Sciences Prague, 165 21, Prague 6, Czech Republic
2 Department of Mathematics, College of Polytechnics Jihlava, Tolstého 16, 586 01, Jihlava, Czech Republic
3 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
4 Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany
5 School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, 2006, Australia
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 479-485.

In [Calc. Var. Partial Differ. Equ. 60 (2021), no. 1, article no. 16 (27 pages)], we revisited a critical Sobolev-type embedding for weighted Sobolev spaces as introduced in [J. Differ. Equations 255 (2013), no. 11, pp. 4312–4336] by Cabré and Ros-Oton. In the same paper, we announced to explore the existence of an extremal function within this framework. In this current work, we not only provide a positive affirmation to this inquiry but extend it to a broader range of weights known as α-homogeneous weights.

Dans le papier [Calc. Var. Partial Differ. Equ. 60 (2021), no. 1, article no. 16 (27 pages)], nous avons revisité une injection de Sobolev critique avec des poids comme introduit dans [J. Differ. Equations 255 (2013), no. 11, pp. 4312–4336] par Cabré et Ros-Oton. Dans le même article, nous avons annoncé d’établir l’existence d’une fonction extrémale sous cette hypothèse. Dans ce papier, nous présentons un résultat affirmatif et nous affaiblissons nos hypothèses.

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DOI : 10.5802/crmath.718
Classification : 46E35, 46E30, 35A23, 26D10
Keywords: Trudinger–Moser inequality, critical Sobolev embedding, Orlicz exponential space, α-homogeneous weight, extremal function
Mots-clés : Inégalité de Trudinger–Moser, injection de Sobolev critique, espace d’Orlicz exponentiel, poids homogènes d’ordre α, fonction extrémale

Petr Gurka 1, 2, 3 ; Daniel Hauer 4, 5

1 Department of Mathematics, Czech University of Life Sciences Prague, 165 21, Prague 6, Czech Republic
2 Department of Mathematics, College of Polytechnics Jihlava, Tolstého 16, 586 01, Jihlava, Czech Republic
3 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
4 Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany
5 School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, 2006, Australia
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Petr Gurka and Daniel Hauer},
     title = {Existence of an extremal function of critical {Sobolev} embedding with an $\alpha $-homogeneous weight},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {479--485},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.718},
     language = {en},
}
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Petr Gurka; Daniel Hauer. Existence of an extremal function of critical Sobolev embedding with an $\alpha $-homogeneous weight. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 479-485. doi : 10.5802/crmath.718. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.718/

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