On the critical points of semi-stable solutions on convex domains of Riemannian surfaces

In this paper we consider semilinear equations $-\mathrm{\Delta }u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has exactly one non-degenerate critical point (a maximum). The proof consists in relating the critical points of the solution with the critical points of a suitable auxiliary function, jointly with a topological degree argument.