Nonlinear fractional Laplacian problems with nonlocal "gradient terms"

2018/12/02 by Abdellaoui, Boumediene, Fernández, Antonio J. · 1 citation
#35B65 #35J62 #35R09 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.00414

Abstract

Let Ω⊂ ℝ^N, N ≥ 2, be a smooth bounded domain. For s ∈ (1/2,1), we consider a problem of the form \\beginaligned (-Δ)^s u amp; = μ(x) \mathbbD_s²(u) + λf(x) , amp; in Ω,
u amp; = 0 , amp; in ℝ^N ∖ Ω, \endaligned . where λ> 0 is a real parameter, f belongs to a suitable Lebesgue space, μ∈ L^∞(Ω) and \mathbbD_s² is a nonlocal "gradient square" term given by \mathbbD_s² (u) = \fraca_N,s2p.v. ∫_ℝ^N \frac|u(x)-u(y)|²|x-y|^N+2s dy . 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.

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