What is the expression of the (non $u \equiv 0$) solutions to
\begin{align\*}
(-\Delta)^s u &= 0 && x \in B\_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B\_r(0),
\end{align\*}
where $$
(-\Delta)^s u(x) = \int\_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}} dy,
$$ ($0<s<1$)
is the fractional Laplacian?



Solution of the fractional Laplace equation on a ball
*Martin kernel* and *Martin representation* is what you are after. Positive solutions are:
$$ u(x) = \int\_{\partial B\_r} \frac{(1 - |x|^2)^s}{|x - y|^N} \, \mu(dy) $$
for any positive measure $\mu$. Signed solution can also be of that form with signed $\mu$, but there are other (more singular) solutions, too.


In other words: take a classical harmonic function $v$ and multiply it by $(1 - |x|^2)^{s-1}$ to get a $(2s)$-harmonic function $u$.


This result is due to Hmissi:


* F. Hmissi, *Fonctions harmoniques pour les potentiels de Riesz sur la boule unité*, Expo. Math. 12(3) (1994), 281–288.


