Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?

Here it is:

Proposition:Let $\alpha \in (0, \infty)$, $p \in (-n, \alpha)$, and $$ f(x) = |x|^p . $$ Then $(-\Delta)^{\alpha/2} f(x)$ is well-defined for $x \ne 0$, and $$ (-\Delta)^{\alpha/2} f(x) = 2^\alpha \frac{\Gamma(\frac{p+n}{2}) \Gamma(\frac{\alpha-p}{2})}{\Gamma(\frac{p+n-\alpha}{2}) \Gamma(-\tfrac{p}{2})} \, |x|^{p - \alpha} . $$ Here we understand the right-hand side is zero if $\tfrac{p+n-\alpha}{2}$ or $-\tfrac{p}{2}$ is a non-positive integer.

This is sort of standard, as it follows from early work of M. Riesz on what is now known as the Riesz potential kernel. Sample reference is the first entry in Table 1 in my survey:

M. Kwaśnicki,

Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko,Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019, DOI:10.1515/9783110571622-007

but there are obviously older sources.

Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko,Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019], DOI:10.1515/9783110571622-007 for a rigorous statement and discussion. $\endgroup$ – Mateusz Kwaśnicki May 22 at 11:32