A note on the Nonlinear Schrödinger Equation on Cartan-Hadamard manifolds with unbounded and vanishing potentials

We study the semilinear equation (Formula presented.) on a Cartan-Hadamard manifold (Formula presented.) of dimension (Formula presented.), and we prove the existence of a nontrivial solution under suitable assumptions on the potential function (Formula presented.). In particular, the decay of (Formula presented.) at infinity is allowed, with some restrictions related to the geometry of (Formula presented.). We generalize some results proved in (Formula presented.) by Alves et al.