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 ω: 1 $$ \left\{ \begin{array}{rcll} (-{\Delta})^{s}u&=&|\nabla u|^{q} + \omega \quad \text{in }\mathbb{R}^{n}, s \in (1/2, 1) u & > &0 \quad \text{in } \mathbb{R}^{n} \lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right. $$ under suitable assumptions on q and ω. Roughly speaking, the condition for existence states that if the measure data is locally controlled by the Riesz fractional capacity, then there is a global solution for the Problem (1). 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.