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Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

Part of: Partial differential equations Miscellaneous topics - Partial differential equations Integral, integro-differential, and pseudodifferential operators Elliptic equations and systems

Published online by Cambridge University Press:  04 February 2020

Boumediene Abdellaoui  and
Antonio J. Fernández
Boumediene Abdellaoui
Affiliation:
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen13000, Algeria (boumediene.abdellaoui@inv.uam.es)
Antonio J. Fernández
Affiliation:
Univ. Polytechnique Hauts-de-France, EA 4015 - LAMAV - FR CNRS 2956, F-59313Valenciennes, France Laboratoire de Mathématiques (UMR 6623), Université de Bourgogne-Franche-Comté, 16 route de Gray, 25030Besançon Cedex, France (antonio_jesus.fernandez_sanchez@univ-fcomte.fr)
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Abstract

Let$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form

$$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$
where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, $\mu \in L^{\infty}$ and $\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by
$$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$
 Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

Keywords

Fractional LaplacianNonlocal gradientCalderón–ZygmundRegularityFractional Poisson equation

MSC classification

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 35J62: Quasilinear elliptic equations 35R09: Integro-partial differential equations 47G20: Integro-differential operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2682 - 2718
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Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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