A note on the NLS equation on Cartan-Hadamard manifolds with unbounded and vanishing potentials

We study the semilinear equation $-{\mathrm{\Delta }}_{g}u+V(\sigma )u=f(u)$ on a Cartan-Hadamard manifold $\mathcal{M}$ of dimension $N\ge 3$, and we prove the existence of a nontrivial solution under suitable assumptions on the potential function $V\in C(\mathcal{M})$. In particular, the decay of $V$ at infinity is allowed, with some restrictions related to the geometry of $\mathcal{M}$. We generalize some results proved in ${\mathbb{R}}^N$ by Alves \emph{et al.}