  We investigate dyonic black hole and dyon solutions of four-dimensional
$SU(N)$ Einstein-Yang-Mills theory with a negative cosmological constant. We
derive a set of field equations in this case, and prove the existence of
non-trivial solutions to these equations for any integer $N$, with $2N-2$ gauge
degrees of freedom. We do this by showing that solutions exist locally at
infinity, and at the event horizon for black holes and the origin for solitons.
We then prove that we can patch these solutions together regularly into global
solutions that can be integrated arbitrarily far into the asymptotic regime.
Our main result is to show that dyonic solutions exist in open sets in the
parameter space, and hence that we can find non-trivial dyonic solutions in a
number of regimes whose magnetic gauge fields have no zeroes, which is likely
important to the stability of the solutions.
