  We study the obstacle problem related to a wide class of nonlinear
integro-differential operators, whose model is the fractional subLaplacian in
the Heisenberg group. We prove both the existence and uniqueness of the
solution, and that solutions inherit regularity properties of the obstacle such
as boundedness, continuity and H\"older continuity up to the boundary. We also
prove some independent properties of weak supersolutions to the class of
problems we are dealing with. Armed with the aforementioned results, we finally
investigate the Perron-Wiener-Brelot generalized solution by proving its
existence for very general boundary data.

Analysis of PDEs