  In second-order scalar-tensor theories we study how the Vainshtein mechanism
works in a spherically symmetric background with a matter source. In the
presence of the field coupling $F(\phi)=e^{-2Q\phi}$ with the Ricci scalar $R$
we generally derive the Vainshtein radius within which the General Relativistic
behavior is recovered even for the coupling $Q$ of the order of unity. Our
analysis covers the models such as the extended Galileon and Brans-Dicke
theories with a dilatonic field self-interaction. We show that, if these models
are responsible for the cosmic acceleration today, the corrections to
gravitational potentials are generally small enough to be compatible with local
gravity constraints.
