Some remarks about the summability of nonlocal nonlinear problems

Abstract

In this note, we will study the problem (-Δ)psu = f(x) on Ω, u = 0 in ℝN∖Ω, where 0 < s < 1, (-Δ)ps is the nonlocal p-Laplacian defined below, Ω is a smooth bounded domain. The main point studied in this work is to prove, adapting the techniques used in [`Basic estimates for solution of elliptic and parabolic equations for a class of nonlocal operators', Discrete Contin. Dyn. Syst., to appear] for the case p = 2 to the general case p ∈ (1, +∞), the summability of the finite energy solutions in terms of the summability of a source term f(x). The aim of this note is to present the results in a way as elementary as possible.

Keywords

• p-fractional laplacian
• stampacchia's method
• moser's method
• calderón–zygmund summability result
• 49j35
• 35a15
• 35s15
• 47g20
• 42b10
• 46e35