  We study existence, stability, and dynamics of linear and nonlinear
stationary modes propagating in radially symmetric multi-core waveguides with
balanced gain and loss. We demonstrate that, in general, the system can be
reduced to an effective ${\cal PT}$-symmetric dimer with asymmetric coupling.
In the linear case, we find that there exist two modes with real propagation
constants before an onset of the ${\cal PT}$-symmetry breaking while other
modes have always the propagation constants with nonzero imaginary parts. This
leads to a stable (unstable) propagation of the modes when gain is localized in
the core (ring) of the waveguiding structure. In the case of nonlinear
response, we show that an interplay between nonlinearity, gain, and loss
induces a high degree of instability, with only small windows in the parameter
space where quasi-stable propagation is observed. We propose a novel
stabilization mechanism based on a periodic modulation of both gain and loss
along the propagation direction that allows bounded light propagation in the
multi-core waveguiding structures.
