Comptes Rendus
Partial differential equations
A note on the global regularity results for strongly nonhomogeneous p,q-fractional problems and applications
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 809-817.

In this article, we communicate with the glimpse of the proofs of new global regularity results for weak solutions to a class of problems involving fractional (p,q)-Laplacian, denoted by (-Δ) p s 1 +(-Δ) q s 2 , for s 2 ,s 1 (0,1) and 1<p,q<. We also obtain the boundary Hölder continuity results for the weak solutions to the corresponding problems involving at most critical growth nonlinearities. These results are almost optimal. Moreover, we establish new Hopf type maximum principle and strong comparison principle. As an application to these new results, we prove the Sobolev versus Hölder minimizer type result, which provides the multiplicity of solutions in the spirit of seminal work [2].

Dans cette note, nous présentons de nouveaux résultats de régularité Höldérienne des solutions faibles d’une classe de problèmes faisant intervenir des opérateurs de diffusion fractionnaire non linéaires et non homogènes de la forme (-Δ) p s 1 +(-Δ) q s 2 avec s 2 ,s 1 (0,1) et 1<p,q<. Précisément, nous obtenons des résultats de régularité intérieure et près du bord pour les solutions faibles de ces problèmes alors que la nonlinéarité du membre de droite est de croissance critique au sens de l’injection de Sobolev. Ce résultat étend les principaux résultats de régularité intérieure de [1] où le cas de l’opérateur homogène (-Δ) p s 1 est étudié, améliore de façon optimale et complète ceux de [8].

Nous établissons par ailleurs un lemme de Hopf et un principe de comparaison fort pour cette classe de problèmes. Nous appliquons ensuite ces résultats pour démontrer la propriété que les minimiseurs locaux de l’énergie associée dans C α (Ω ¯) avec α(0,s 1 ) sont aussi minimiseurs locaux dans W 0 s 1 ,p (Ω) dans l’esprit de l’article pionnier [2]. Ceci conduit à des nouveaux résultats de muliplicité de solutions pour ces problèmes non locaux et fortement non homogènes.

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Accepted:
Published online:
DOI: 10.5802/crmath.344
Classification: 35J60, 35R11, 35B45, 35D30

Jacques Giacomoni 1; Deepak Kumar 2; Konijeti Sreenadh 2

1 Université de Pau et des Pays de l’Adour, LMAP (UMR E2S-UPPA CNRS 5142), Bat. IPRA, Avenue de l’Université F-64013 Pau, France
2 Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-110016, India
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {A note on the global regularity results for strongly nonhomogeneous $p,q$-fractional problems and applications},
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Jacques Giacomoni; Deepak Kumar; Konijeti Sreenadh. A note on the global regularity results for strongly nonhomogeneous $p,q$-fractional problems and applications. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 809-817. doi : 10.5802/crmath.344. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.344/

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