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

PABLO OCHOA; MARIA LAURA DE BORBÓN

POTENTIAL ANALYSIS

SPRINGER

Lugar: Berlin; Año: 2021 vol. 55 p. 677 - 698

0926-2601

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 $omega$:egin{equation}left{egin{array}{rcll} (-Delta)^su&=&|abla u|^q + omega quad extnormal{in }mathbb{R}^n,, ,,s in (1/2, 1) u & > &0 quad ext{in } mathbb{R}^{n} lim_{|x|o infty}u(x) & =& 0,end{array}ight.end{equation}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.