No-scalar-hair  theorems for nonminimally coupled fields with quartic self-interaction

Self-gravitating scalar fields with nonminimal coupling to gravity and having a quartic self-interaction are considered in the domain of outer communications of a static black hole. It is shown that there is no value of the nonminimal coupling parameter for which nontrivial static black hole solutions exist. This result establishes the correctness of Bekenstein ``no-scalar-hair'' conjecture for quartic self-interactions.