  In this paper we discuss the existence and non--existence of weak solutions
to parametric equations involving the Laplace-Beltrami operator $\Delta_g$ in a
complete non-compact $d$--dimensional ($d\geq 3$) Riemannian manifold
$(\mathcal{M},g)$ with asymptotically non--negative Ricci curvature and
intrinsic metric $d_g$. Namely, our simple model is the following problem $$
\left\{ \begin{array}{ll} -\Delta_gw+V(\sigma)w=\lambda \alpha(\sigma)f(w) &
\mbox{ in } \mathcal{M}\\ w\geq 0 & \mbox{ in } \mathcal{M} \end{array}\right.
$$ where $V$ is a positive coercive potential, $\alpha$ is a positive bounded
function, $\lambda$ is a real parameter and $f$ is a suitable continuous
nonlinear term. The existence of at least two non--trivial bounded weak
solutions is established for large value of the parameter $\lambda$ requiring
that the nonlinear term $f$ is non--trivial, continuous, superlinear at zero
and sublinear at infinity. Our approach is based on variational methods. No
assumptions on the sectional curvature, as well as symmetry theoretical
arguments, are requested in our approach.
