Penrose hypothesis and instability of naked singularities in static spherically symmetric systems with scalar fields

General relativistic static spherically symmetric (SSS) asymptotically flat configurations with scalar fields typically contain naked singularities at the center. We consider minimally coupled scalar fields with power-law potentials leading to the Coulomb asymptotic of the field $\phi(r)\approx Q/r$ for large values of the radial variable r. The configurations are uniquely defined by total mass and a Q-parameter characterizing the strength of the scalar field at spatial infinity. The focus is on the linear stability against radial (monopole) perturbations of the SSS configurations satisfying conditions of asymptotic flatness. Our numerical investigations show the existence of divergent modes of small perturbations against the static background, at least for sufficiently small values of Q. This means instability of the configurations, confirming the well-known Penrose conjecture about the nonexistence of naked singularities - in this particular case. On the other hand, we have not found divergent modes of linear radial perturbations for sufficiently large Q.