  We prove that, for a generic set of smooth prescription functions $h$ on a
closed ambient manifold, there always exists a nontrivial, smooth, closed
hypersurface of prescribed mean curvature $h$. The solution is either an
embedded minimal hypersurface with integer multiplicity, or a non-minimal
almost embedded hypersurface of multiplicity one.
  More precisely, we show that our previous min-max theory, developed for
constant mean curvature hypersurfaces, can be extended to construct min-max
prescribed mean curvature hypersurfaces for certain classes of prescription
function, including smooth Morse functions and nonzero analytic functions. In
particular we do not need to assume that $h$ has a sign.
