  We study $L^{p}\times L^{q}\rightarrow L^{r}$-boundedness of (sub)bilinear
maximal functions associated with degenerate hypersurfaces. First, we obtain
the maximal bound on the sharp range of exponents $p,q,r$ (except some border
line cases) for the bilinear maximal functions given by the model surface
$\big\{(y,z)\in\mathbb{R}^{n}\times
\mathbb{R}^{n}:|y|^{l_{1}}+|z|^{l_{2}}=1\big\}$, $(l_{1},l_{2})\in
[1,\infty)^2$, $n\ge 2$.
  Our result manifests that nonvanishing Gaussian curvature is not good enough,
in contrast with $L^p$-boundedness of the (sub)linear maximal operator
associated to hypersurfaces, to characterize the best possible maximal
boundedness. Secondly, we consider the bilinear maximal function associated to
the finite type curve in $\mathbb R^2$ and obtain a complete characterization
of the maximal bound. We also prove multilinear generalizations of the
aforementioned results.
