Schr\"odinger equation on Cartan-Hadamard manifolds with oscillating nonlinearities

TL;DR

This paper proves the existence of infinitely many solutions to a nonlinear Schrödinger equation on Cartan-Hadamard manifolds with oscillating nonlinearities, without relying on topological index theory.

Contribution

It introduces a novel approach to establish solution existence for oscillating nonlinearities on Cartan-Hadamard manifolds, bypassing traditional topological methods.

Findings

Existence of infinitely many solutions established

Applicable to nonlinearities oscillating at zero or infinity

No use of topological index theory required

Abstract

We study the equation $-\Delta_g w+w=\lambda \alpha(\sigma) f(w)$ on a $d$-dimensional homogeneous Cartan-Hadamard Manifold $\mathcal{M}$ with $d \geq 3$. 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.