Elliptic problems on complete non-compact Riemannian manifolds with asymptotically non-negative Ricci curvature

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.