We consider an augmented Einstein-Maxwell-scalar model including an axionic-type coupling between the scalar and electromagnetic field. We study dyonic black hole solutions in this model. For the canonical axionic coupling emerging from high energy physics, all charged black holes have axion hair. We present their domain of existence and investigate some physical properties. For other axionic-type couplings, two classes of black hole solutions may co-exist in the model: scalar-free Reissner-Nordström black holes and scalarised black holes. We show that in some region of the parameter space, the scalar-free solutions are unstable. Then, there is non-uniqueness since new scalarised black hole solutions with the same global charges also exist, which are entropically preferred over the scalar-free solutions and, moreover, emerge dynamically from the instability of the former. 1 pedrogsilvafernandes@tecnico.ulisboa.pt 2 carlosherdeiro@tecnico.ulisboa.pt 3 pomboalexandremira@ua.pt 4 eugen.radu@ua.pt 5 nicolas.sanchis@tecnico.ulisboa.pt 1 arXiv:1908.00037v1 [gr-qc] 31 Jul 2019Three types of bosonic fields, each consistent on its own as a classical relativistic field theory, are used in the physical description of Nature: spin 0, 1 and 2 fields. According to the accepted models, the last two fields are realised, in their massless version, as the electromagnetic and gravitational field, whereas the scalar field is realised, in a massive, self-interacting version, as the Higgs boson. Beyond these concrete realisations, it has been a recurrent speculation in theoretical physics, ranging from quantum gravity to cosmology, that scalar fields play other roles. One particular example is a pseudo-scalar field known as the axion. This (yet unobserved) particle was suggested in order to solve the strong CP problem, by Peccei and Quinn [1] (see also [2][3][4]). But it was later understood that besides fulfilling its original purpose, the axion could have far reaching implications in cosmology, namely as a candidate for dark matter [5][6][7][8]. More recently, ultralight axion-like fields have been suggested to arise naturally from string theory compactifications [9,10], providing yet another possible origin in the context of a fundamental theory. The theoretical soundness of axions motivated experiments, both proposed and conducted, to detect axionic imprints -see e.g. [11][12][13].In the context of strong gravity, namely black hole (BH) spacetimes, the interplay between the gravitational, electromagnetic and non-standard model scalar fields has a long history. If the scalar field is minimally coupled to the electromagnetic and gravitational field, remarkably, it cannot exist as an equilibrium configuration around spherical BHs [14]. This is a realisation of the celebrated "no-hair" property of BHs -see, e.g. [15,16]. If non-minimal couplings are allowed, however, hairy BH solutions exist. In this paper we shall be interested in such BHs when an axionic-type non-minimal coupling between the scalar and the electromagnetic field are ...