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

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 $ω$: $\left\{ \begin{array}{rcll} (-Δ)^su&=&|\nabla u|^q + ω\quad \text{in }\mathbb{R}^n,\, \,\,s \in (1/2, 1)\\u & > &0 \quad \text{in } \mathbb{R}^{n}\\\lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right.$under suitable assumptions on $q$ and $ω$. 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 $ω$ 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 $ω$ are also given in different function spaces.

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