  We prove the existence of ground state solutions for a class of nonlinear
elliptic equations, arising in the production of standing wave solutions to an
associated family of nonlinear Schr\"odinger equations. We examine two
constrained minimization problems, which give rise to such solutions. One
yields what we call $F_\lambda$-minimizers, the other energy minimizers. We
produce such ground state solutions on a class of Riemannian manifolds called
weakly homogeneous spaces, and establish smoothness, positivity, and decay
properties. We also identify classes of Riemannian manifolds with no such
minimizers, and classes for which essential uniqueness of positive solutions to
the associated elliptic PDE fails.
