A note on quasilinear equations with fractional diffusion
2020
In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \begin{eqnarray*}
\left\lbrace
\begin{array}{l}
(-\Delta)^{s}u + |\nabla u|^{p} =f \quad\text{ in } \Omega
\qquad \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 $p<2s$, that is the natural growth, we are able to show the existence for all $f\in L^1(O)$. In the subcritical case, that is, for $p < p_{*}:=N/(N-2s+1)$, 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.
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