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 [Formula presented] in a complete non-compact [Formula presented]-dimensional ([Formula presented]) Riemannian manifold [Formula presented] with asymptotically non-negative Ricci curvature and intrinsic metric [Formula presented]. Namely, our simple model is the following problem [Formula presented]where [Formula presented] is a positive coercive potential, [Formula presented] is a positive bounded function, [Formula presented] is a real parameter and [Formula presented] 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 [Formula presented] requiring that the nonlinear term [Formula presented] 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.