# A characterization related to Schr\"odinger equations on Riemannian manifolds

## TL;DR

This paper investigates the existence of solutions to a nonlinear Schrödinger equation on non-compact Riemannian manifolds with asymptotically non-negative Ricci curvature, using variational methods to establish a characterization criterion.

## Contribution

It provides a new characterization criterion for the existence of solutions to Schrödinger equations on Riemannian manifolds with specific geometric conditions.

## Key findings

- Existence of solutions depends on the parameter mbda and the potential V.
- Variational methods effectively establish solution criteria.
- Results extend understanding of Schrödinger equations on curved spaces.

## Abstract

In this paper we consider the following problem $$\begin{cases} -\Delta_{g}u+V(x)u=\lambda\alpha(x)f(u), & \mbox{in }M\\ u\geq0, & \mbox{in }M\\ u\to0, & \mbox{as }d_{g}(x_{0},x)\to\infty \end{cases}$$where $(M,g)$ is a $N$-dimensional ($N\geq3)$, non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, $\lambda$ is a real parameter, $V$ is a positive coercive potential, $\alpha$ is a bounded function and $f$ is a suitable nonlinearity. By using variational methods we prove a characterization result for existence of solutions for our problem.