A capacity-based condition for existence of solutions to fractional elliptic equations with first-order terms and measures

Abstract

In this manuscript, we appeal to Potential Theory to provide a sufficient condition for existence of distributional solutions to fractional elliptic problems with non-linear first-order terms and measure data $\omega$: $$ \left\{ \begin{array}{rcll} (-\Delta)^su&=&|\nabla u|^q + \omega \quad \text{in }\mathbb{R}^n,\, \,\,s \in (1/2, 1)\\u & > &0 \quad \text{in } \mathbb{R}^{n}\\\lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right. $$under suitable assumptions on $q$ and $\omega$. Roughly speaking, the condition for exis\-tence states that if the measure data is locally controlled by the Riesz fractional capacity, then there is a global solution for the equation. We also show that if a positive solution exists, necessarily the measure $\omega$ will be absolutely continuous with respect to the associated Riesz capacity, which gives a partial reciprocal of the main result of this work. Finally, estimates of $u$ in terms of $\omega$ are also given in different function spaces.