The paper is concerned with the scalar curvature equation with prescribed mean curvature on the boundary of a given Riemannian manifold. Just as in \textit{T. Ouyang} [Trans. Am. Math. Soc. 331, No. 2, 503-527 (1992; Zbl 0759.35021)], this Riemannian manifold is assumed to have negative constant scalar curvature in the interior and zero mean curvature on the boundary. By making a good use of bifurcation theory from a simple eigenvalue, the author shows us that the problem with prescribed nonpositive scalar curvature and nonpositive mean curvature is not always solvable.