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 positive solutions to the semilinear elliptic equation $$-{\Delta }_{g}w+Vw=\alpha f(w)+\lambda win\mathcal{M}.$$ The potential $V:\mathcal{M}\to \mathbb{R}$ 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 solution exists as soon as the parameter $\lambda$ is sufficiently close to an eigenvalue of the operator $-{\Delta }_g$.