A note on quasilinear equations with fractional diffusion

IRENEO PERAL; PABLO OCHOA; BOUMEDIENE ABDELLAOUI

Mathematics in Engineering

AIMS Press

Año: 2020

In this paper, we study the existence of distributional solutions of the following non-local elliptic problem􏰋 (−∆)su+|∇u|p = f inΩu = 0 inRNΩ, s∈(1/2,1).We are interested in the relation between the regularity of the source term f , and the regularity of the corresponding solution. If 1 < p < 2s, that is the natural growth, we are able to show the existence for all f ∈ L1(Ω).In the subcritical case, that is, for 1 < p < p∗ := N/(N − 2s + 1), we show that solutions are C1,α forf ∈ Lm, with m large enough. In the general case, we achieve the same result under a condition on the size of the source. As an application, we may show that for regular sources, distributional solutions are viscosity solutions, and conversely.