Multiple solutions for Schr\"odinger equations on Riemannian manifolds via $\nabla$-theorems

TL;DR

This paper proves the existence of multiple positive solutions to a Schrödinger-type equation on non-compact Riemannian manifolds using $ abla$-theorems, especially near eigenvalues of the Laplace operator.

Contribution

It introduces a novel application of $ abla$-theorems to establish multiple solutions for Schrödinger equations on Riemannian manifolds, extending previous methods.

Findings

At least three solutions exist near eigenvalues of the Laplacian.

The results apply to manifolds with coercive potentials and subcritical nonlinearities.

The approach broadens the scope of solution existence results for elliptic equations on manifolds.

Abstract

We consider a smooth, complete and non-compact Riemannian manifold $(\mathcal{M},g)$ of dimension $d \geq 3$, and we look for positive solutions to the semilinear elliptic equation $$ -\Delta_g w + V w = \alpha f(w) + \lambda w \quad\hbox{in $\mathcal{M}$}. $$ The potential $V \colon \mathcal{M} \to \mathbb{R}$ 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 $-\Delta_g$.