Existence Serrin type results for the Dirichlet problem for the prescribed mean curvature equation in Riemannian manifolds

Given a complete $n$-dimensional Riemannian manifold $M$, we study the existence of vertical graphs in $M\times \mathbb{R}$ with prescribed mean curvature $H=H(x,z)$. Precisely, we prove that the Dirichlet problem for the vertical mean curvature equation in a smooth bounded domain $\mathrm{\Omega }\subset M$ has solution for arbitrary smooth boundary data if $(n-1)$\mathcal{H}${\partial\Omega}(y)\geq n\sup\limits{z\in$\mathbb{R}$}\left|{H(y,z)}\right|$foreach$y\in\partial\Omega $providedthefunction$H$alsosatisfies$$\mathrm{R}\mathrm{i}\mathrm{c}\mathrm{c}$x\geq n\sup\limits{z\in$\mathbb{R}$}\left|\nabla_x H(x,z)\right|-\dfrac{n^2}{n-1}\inf\limits_{z\in$\mathbb{R}$}\left(H(x,z)\right)^2$foreach$x\in\Omega$\mathrm{.}Inthecasewhere$M=$\mathbb{H}$^n$ we also establish an existence result if the condition $\underset{\mathrm{\Omega }\times \mathbb{R}}{\sup }\mid H(x,z)\mid \le \frac{n-1}{n}$ holds in the place of the condition involving the Ricci curvature. Finally, we have a related result when $M$ is a Hadamard manifold whose sectional curvature $K$ satisfies $-c^2\le K\le -1$ for some $c>1$. We generalize a classical result of Serrin when the ambient is the Euclidean space.