  Small instantons can play an important role in Yang-Mills theories whose
gauge couplings are sizeable at small distances. An interesting class of
theories where this could occur is in weakly-gauged holographic models (dual to
Yang-Mills theories interacting with strongly-coupled CFTs), since gauge
couplings are indeed enhanced towards the UV boundary of the 5D AdS space.
However, contrary to expectations, we show that small instantons in these
non-asymptotically-free models are highly suppressed and ineffective. This is
due to the conservation of topological charge that forbids instantons to be
localized near the UV boundary. Despite this fact we find non-trivial UV
localized instanton-anti-instanton solutions of the Yang-Mills equations where
the topological charges annihilate in the AdS bulk. These analytic solutions
arise from a 5D conformal transformation of the uplifted 4D instanton. Our
analysis therefore reveals unexpected nonperturbative configurations of
Yang-Mills theories when they interact with strongly-coupled CFTs.
