A note on quasilinear equations with fractional diffusion

Abstract

In this paper, we study the existence of distributional solutions of the following non-local elliptic problem (equation presented). 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 ∈ L 1 (Ω). In the subcritical case, that is, for 1 < p < p∗ := N/(N − 2s + 1), we show that solutions are C 1,α for f ∈ L m , 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