In this talk we study the existence of $n$-dimensional hypersurface with $C^1$ mean curvature function in warped product manifolds under barrier conditions.  With those barrier conditions we construct a family of graphs satisfying auxiliary mean curvature equations.  These graphs will converge to an almost minimal boundary in the sense of varifold. Applying the regularity of the almost minimal boundary,  this limit is smooth, embedded satisfying prescribed mean curvature when $n/leq 6$. Similar schemes also work in warped product manifolds and  cylinders.

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