  We are interested in regularity properties of semi-stable solutions for a
class of singular semilinear elliptic problems with advection term defined on a
smooth bounded domain of a complete Riemannian manifold with zero Dirichlet
boundary condition. We prove uniform Lebesgue estimates and we determine the
critical dimensions for these problems with nonlinearities of the type Gelfand,
MEMS and power case. As an application, we show that extremal solutions are
classical whenever the dimension of the manifold is below the critical
dimension of the associated problem. Moreover, we analyze the branch of minimal
solutions and we prove multiplicity results when the parameter is close to
critical threshold and we obtain uniqueness on it. Furthermore, for the case of
Riemannian models we study properties of radial symmetry and monotonicity for
semi-stable solutions.
