  In this paper, we deal with three aspects of $p$-monotone operators. First we
study $p$-monotone operators with a unique maximal extension (called
pre-maximal), and with convex graph. We then deal with linear operators, and
provide characterizations of $p$-monotonicity and maximal $p$-monotonicity.
Finally we show that the Brezis-Browder theorem preserves $p$-monotonicity in
reflexive Banach spaces.
