  It was recently found that in certain flavours of scalar-Gauss-Bonnet gravity
linearly stable bald black holes can co-exist with stable scalarized solutions.
The transition between both can be ignited by a large nonlinear perturbation,
thus the process was dubbed non-linear scalarization, and it happens with a
jump that leads to interesting astrophysical implications. Generalizing these
results to the case of nonzero scalar field potential is important because a
massive self-interacting scalar field can have interesting theoretical and
observational consequences, e.g. reconcile scalar-Gauss-Bonnet gravity with
binary pulsar observation, stabilize black hole solutions, etc. That is why in
the present paper, we address this open problem. We pay special attention to
the influence of a scalar field mass and self-interaction on the existence of
scalarized phases and the presence of a jump between stable bald and hairy back
holes. Our results show that both the addition of a mass and positive
self-interaction of the scalar field result in suppression or quenching of the
overall scalarization phenomena. A negative scalar field self-interaction
results in an increase of the scalarization. The presence and the size of the
jump, though, are not so sensitive to the scalar field potential.
