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 $\omega$: $$\{\begin{array}{rcl}(-\Delta {)}^{s}u& =& |\nabla u{|}^q+\omega \text{in }{\mathbb{R}}^n,s\in (1/2,1)\\ u& >& 0\text{in }{\mathbb{R}}^n\\ {\lim }_{|x|\to \infty }u(x)& =& 0,\end{array}$$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.