  We study a fully nonlinear PDE involving a linear combination of symmetric
polynomials of the K\"ahler form on a K\"ahler manifold. A $C^0$ \emph{a
priori} estimate is proven in general and a gradient estimate is proven in
certain cases. Independently, we also provide a method-of-continuity proof via
a path of K\"ahler metrics to recover the existence of solutions in some of the
known cases. Known results are then applied to an analytic problem arising from
Chern-Weil theory and to a special Lagrangian-type equation arising from mirror
symmetry.
