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Article Dans Une Revue Journal of Differential Equations Année : 2021

Hardy-Sobolev inequalities with singularities on non smooth boundary. Part 2: Influence of the global geometry in small dimensions

Résumé

We consider Hardy-Sobolev nonlinear equations on domains with singularities. We introduced this problem in Cheikh-Ali [4]. Under a local geometric hypothesis, namely that the generalized mean curvature is negative (see (6) below), we proved the existence of extremals for the relevant Hardy-Sobolev inequality for large dimensions. In the present paper, we tackle the question of small dimensions that was left open. We introduce a "mass", that is a global quantity, the positivity of which ensures the existence of extremals in small dimensions. As a byproduct, we prove the existence of solutions to a perturbation of the initial equation via the Mountain Pass Lemma.
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Dates et versions

hal-02048901 , version 1 (25-02-2019)
hal-02048901 , version 2 (05-10-2020)

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Citer

Hussein Cheikh Ali. Hardy-Sobolev inequalities with singularities on non smooth boundary. Part 2: Influence of the global geometry in small dimensions. Journal of Differential Equations, 2021, 270, pp.185-216. ⟨10.1016/j.jde.2020.07.043⟩. ⟨hal-02048901v2⟩
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