A note on quasilinear equations with fractional diffusion

In this paper, we study the existence of distributional solutions of the following non-local elliptic problem $ \begin{eqnarray*} \left\lbrace \begin{array}{lll} (-\Delta)^{s}u + |\nabla u|^{p} & = & f \quad\text{ in } \Omega\\ \qquad \qquad \,\,\,\,\,\:\: u & = & 0 \,\,\,\,\,\,\,\text{ in } \mathbb{R}^{N}\setminus \Omega, \quad s \in (1/2, 1). \\ \end{array} \right. \end{eqnarray*} $ 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 $f\in L^1(\Omega)$. In the subcritical case, that is, for $1 < p < p_{*}: = N/(N-2s+1)$, we show that solutions are $\mathcal{C}^{1, \alpha}$ for $f \in L^{m}$, 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.