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Vainshtein mechanism in second-order scalar-tensor theories

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 $F(ϕ)=e^{-2Qϕ}$ 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.