Schrödinger equation on Cartan-Hadamard manifolds with oscillating nonlinearities

We study the equation $-{\mathrm{\Delta }}_{g}w+w=\lambda \alpha (\sigma )f(w)$ on a $d$-dimensional homogeneous Cartan-Hadamard Manifold $\mathcal{M}$ with $d\ge 3$. Without using the theory of topological indices, we prove the existence of infinitely many solutions for a class of nonlinearities $f$ which have an oscillating behavior either at zero or at infinity.