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Article Dans Une Revue Proceedings of the Royal Society of Edinburgh: Section A, Mathematics Année : 2020

Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

Résumé

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.

Dates et versions

hal-03337289 , version 1 (07-09-2021)

Identifiants

Citer

Boumediene Abdellaoui, Antonio J. Fernández Sánchez. Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2020, 150 (5), pp.2682-2718. ⟨10.1017/prm.2019.60⟩. ⟨hal-03337289⟩
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