  Previous studies showed that, in the presence of a simple and well-motivated
self-interaction scalar potential, asymptotically flat and spherical black
holes can carry minimally coupled and charged scalar cloud/hair in
Einstein-Maxwell gravity. We extend these studies to Einstein-Born-Infeld
gravity to consider the effect of nonlinearity of the electromagnetic field.
Series of spherical cloudy/hairy black hole solutions are constructed
numerically. Results show that increasing the Born-Infeld coupling constant $b$
will make the domain of existence of the solution shrink or even disappear when
$b$ is large enough. This implies that, competing with the gravitation,
nonlinearity of the electromagnetic field will make the formation of scalar
cloud/hair harder or even impossible.
