A note on the Nonlinear Schr?dinger Equation on Cartan-Hadamard manifolds with unbounded and vanishing potentials

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