  We study linear perturbations against static spherically symmetric background
configurations of General Relativity with a real scalar field (SF), which is
minimally coupled with gravity; it is non-linear due to the presence of the
self-action potential. The background solutions have a naked singularity at the
center of the configuration. The focus is on the stability of the background
and fundamental frequencies of the quasi-normal modes (QNM) of the axial
perturbations in the Regge-Wheeler gauge. The problem is reduced to one
hyperbolic master equation with an effective potential $W_{\rm eff}$, which
turns out to be positive for a general non-negative SF potential; this ensures
the linear stability with respect to this kind of perturbations.
  For numerical simulations, the SF potential was chosen in the power-law form
$V(\phi)\sim\phi^{2n}$ with $2<n\le 40$. We extracted the fundamental
frequencies of QNM for different $n$ and various sets of the background
configuration parameters. The results show that even for a small background SF,
there is a significant difference between the fundamental frequencies and ones
in case of the Schwarzschild background. The results are also compared with the
case of the Fisher-Janis-Newman-Winicour background dealing with a massless
linear scalar field.
