  We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in
$\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic
ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class
of Riemannian manifolds includes the classical \textit{space forms}, i.e., the
Euclidean, elliptic, and hyperbolic spaces. For the class of semistable
solutions we prove radial symmetry and monotonicity. Furthermore, we establish
$L^\infty$, $L^p$, and $W^{1,p}$ estimates which are optimal and do not depend
on the nonlinearity $f$. As an application, under standard assumptions on the
nonlinearity $\lambda f(u)$, we prove that the corresponding extremal solution
$u^*$ is bounded whenever $n\leq9$. To establish the optimality of our
regularity results we find the extremal solution for some exponential and power
nonlinearities using an improved weighted Hardy inequality.
