On some elliptic fractional $s(·)$ problems with singular potential and general datum

Biroud, Kheireddine; Laamri, El-Haj

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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.

Classification : 35B05, 35K15, 35B40, 35K55, 35K65
Keywords: The variable-order fractional linear elliptic problems, singular nonlinearities, capacities, measure decomposition, Harnack inequality

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