Given a complete d-dimensional Riemannian manifold \((\mathcal M,g)\) I will prove that, for any \(p\in\mathcal M), any nonlinearity \(f(q,u)\) with \(f(p,0)>0\) and for any integer \(n\ge2\), there exists a sequence of smooth bounded domains \(\Om_k\subset\mathcal M\) containing p and corresponding positive solutions \(u_k:\Om_k\to\R^+\) to the Dirichlet boundary problem \(\( \begin{cases} -\Delta_g u_k=f(\cdot,u_k) \quad & \mbox{ in }\Om_k\,,\\ u_k=0 & \mbox{ on }\partial\Om_k\,. \end{cases} \)\) such that the solution \(u_k\) have exactly 2n-1 nondegenerate critical points in \(\Om_k\) (specifically, n nondegenerate maxima and n-1 nondegenerate saddles). Moreover the domains \(\Om_k\) are star-shaped with respect to p and become ``nearly geodesically convex'', in a precise sense, as \(k\to+\infty\). The proof relies on similar results in \(\R^d$, $d\geq 3\), for the torsion problem. The talk is based on past and ongoing results involving A. Enciso. and M. Grossi.