$L^{q}$ estimates for nonlocal p-Laplacian type equations with BMO kernel coefficients in divergence form

Abstract

We study s-fractional p-Laplacian type equations with discontinuous kernel coefficients in divergence form to establish W s+σ,q estimates for any choice of pairs (σ, q) with q ∈ (p, ∞) and σ ∈ 0, min s p−1 , 1 − s under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a non-homogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in the linear case.under a possibly discontinuous kernel coefficient A(x, y). For the classical case that p = q and σ = 0, there is a unique weak solution u ∈ W s,p (Ω) ∩ L p−1 sp (R n ) with the standard energy estimate (2.15), as follows from Lemma 2.7 below.We first discuss a motivation of the study (1.1) from the corresponding local problem when s = 1. According to the well-known elliptic theory, for a weak solution w of −div (B|Dw| p−2 Dw) = −div (|F | p−2 F ) in Ω, which comes from the Euler-Lagrange equation of the functional v → ˆΩ 1 p B|Dv| p − |F | p−2 F • Dv dx

The divergence theorem and nonlocal counterparts

“…Using our fractional divergence and gradient, it is thus possible to write the fractional 𝑝-Laplace like the local 𝑝-Laplace Δ 𝑝 𝑢 = div ⋅(|∇𝑢| 𝑝−2 ∇𝑢). Similar representations have been given in other works such as [6,32,34]. → 1 as 𝑠 → 1, a proof of which can be found in several sources, see, for example, [22] for an elegant proof.…”