By virtue of barrier arguments we prove C^\alpha -regularity up to the boundary for the weak solutions of a non-local, non-linear problem driven by the fractional p -Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular ( p < 2 ) of degenerate ( p > 2 ) case.

A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p, q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.