Multiple solutions for Schrödinger equations on Riemannian manifolds via ∇ -theorems

We consider a smooth, complete and non-compact Riemannian manifold (M, g) of dimension d≥ 3 , and we look for solutions to the semilinear elliptic equation -Δgw+V(σ)w=α(σ)f(w)+λwinM.The potential V: M→ R is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of ∇ -theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter λ is sufficiently close to an eigenvalue of the operator - Δg.