Multiple solutions for Schrödinger equations on Riemannian manifolds via \(\nabla \)-theorems

Abstract

We consider a smooth, complete and non-compact Riemannian manifold \((\mathcal {M},g)\) of dimension \(d \ge 3\), and we look for solutions to the semilinear elliptic equation

$$\begin{aligned} -\varDelta _g w + V(\sigma ) w = \alpha (\sigma ) f(w) + \lambda w \quad \hbox {in }\mathcal {M}. \end{aligned}$$

The potential \(V :\mathcal {M} \rightarrow \mathbb {R}\) 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 \(\nabla \)-theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter \(\lambda \) is sufficiently close to an eigenvalue of the operator \(-\varDelta _g\).