On nonlinear Schr\"{o}dinger equations on the hyperbolic space

TL;DR

This paper investigates the existence of weak solutions for nonlinear Schrödinger equations on hyperbolic space, specifically using symmetry and group theory to establish solutions on the Poincaré ball model.

Contribution

It introduces a novel approach combining symmetric criticality and group theory to prove existence of solutions on hyperbolic spaces.

Findings

Existence of nontrivial weak solutions established

Use of Palais principle of symmetric criticality

Application of group theoretical methods

Abstract

We study existence of weak solutions for certain classes of nonlinear Schr\"{o}dinger equations on the Poincar\'{e} ball model $\mathbb{B}^N$, $N\geq 3$. By using the Palais principle of symmetric criticality and suitable group theoretical arguments, we establish the existence of a nontrivial (weak) solution.