We derive exact, asymptotically flat black hole solutions of Einstein-scalar gravity sourced by a nontrivial scalar field with $1/r$ asymptotic behavior. They are determined using an ansatz for the scalar field profile and working out, together with the metric functions, the corresponding form of the scalar self-interaction potential. Near the singularity the black hole behaves as the Janis-Newmann-Winicour-Wyman solution. We also work out a consistent thermodynamical description of our black hole solutions. At large mass our hairy black holes have the same thermodynamical behavior of the Schwarzschild black hole, whereas for small masses they differ substantially from the latter.
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Cadoni et al. (2015) studied this question.
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