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On some elliptic fractional s(·) problems with singular potential and general datum
Kheireddine Biroud1; El-Haj Laamri2
1 Laboratoire d’Analyse Non linéaire et Mathématiques Appliquées, École Supérieure de Management de Tlemcen, No. 01, Rue Barka Ahmed Bouhannak, Imama, Tlemcen 13000, Algeria
2 Institut Élie Cartan de Lorraine, Université de Lorraine, B.P: 239, 54506 Vandœuvre-Lés-Nancy Cedex, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 3, pp. 681-738.

The purpose this work is to address the question of existence and regularity of solutions to a class of nonlocal elliptic problems with variable-order fractional Laplace operator and whose behaviors are complicated by the presence of singular nonlinearities. First, we prove the existence of weak solutions for a large class of data, including measures in some cases. We also obtain additional regularity properties under suitable extra assumptions. Second, we show that, in the case of measures datum, existence analysis is strongly related to the fractional capacity associated to the fractional Sobolev spaces. As a consequence, we get the natural form of the adequate “fractional gradient” when dealing with the Hamilton–Jacobi fractional equation with nonlocal gradient term in the sense of Boccardo–Gallouët–Orsina decomposition Problem.

Le but de ce travail est d’étudier la question de l’existence et la régularité des solutions d’une classe de problèmes elliptiques non locaux gouvernés par l’opérateur de Laplace fractionnaire d’ordre variable, et dont le second membre est non linéaire et comporte des singularités. En premier lieu, nous prouvons l’existence de solutions faibles pour une grande classe de données, y compris pour des données mesures. Ensuite, nous montrons que lorsque les données sont régulières, les solutions le sont aussi. Enfin, nous montrons que, dans le cas de données de mesures, l’existence de solutions est fortement liée à la capacité fractionnaire associée aux espaces de Sobolev fractionnaires. Ce qui nous a permis d’obtenir la forme naturelle du « gradient fractionnaire » adéquat lorsque nous traitons l’équation fractionnaire de Hamilton–Jacobi avec un gradient non local dans le sens de décomposition de Boccardo–Gallouët–Orsina.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1785
Classification: 35B05, 35K15, 35B40, 35K55, 35K65
Keywords: The variable-order fractional linear elliptic problems, singular nonlinearities, capacities, measure decomposition, Harnack inequality

Kheireddine Biroud 1; El-Haj Laamri 2

1 Laboratoire d’Analyse Non linéaire et Mathématiques Appliquées, École Supérieure de Management de Tlemcen, No. 01, Rue Barka Ahmed Bouhannak, Imama, Tlemcen 13000, Algeria
2 Institut Élie Cartan de Lorraine, Université de Lorraine, B.P: 239, 54506 Vandœuvre-Lés-Nancy Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {On some elliptic fractional $s(\,\cdot \,)$ problems with singular potential and general datum},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Kheireddine Biroud; El-Haj Laamri. On some elliptic fractional $s(\,\cdot \,)$ problems with singular potential and general datum. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 3, pp. 681-738. doi : 10.5802/afst.1785. https://afst.centre-mersenne.org/articles/10.5802/afst.1785/

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