# On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space

## TL;DR

This paper investigates nonlinear Helmholtz equations in hyperbolic space, establishing existence of solutions, analyzing their behavior, and providing counterexamples to Strichartz estimates using a new Limiting Absorption Principle.

## Contribution

It introduces a novel Limiting Absorption Principle for the Helmholtz operator in hyperbolic space and constructs solutions to nonlinear Helmholtz equations for all positive parameters.

## Key findings

- Existence of nontrivial solutions for all >0 and p>2.
- Analysis of oscillatory behavior and decay rates of radial solutions.
- Counterexamples to certain Strichartz estimates in hyperbolic space.

## Abstract

In this paper, we study nonlinear Helmholtz equations (NLH) $-\Delta_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -\lambda^2 u = \Gamma|u|^{p-2}u$ in $\mathbb{H}^N$, $N\geq 2$ where $\Delta_{\mathbb{H}^N}$ denotes the Laplace-Beltrami operator in the hyperbolic space $\mathbb{H}^N$ and $\Gamma\in L^\infty(\mathbb{H}^N)$ is chosen suitably. Using fixed point and variational techniques, we find nontrivial solutions to (NLH) for all $\lambda>0$ and $p>2$. The oscillatory behaviour and decay rates of radial solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds and Damek-Ricci spaces. Our results rely on a new Limiting Absorption Principle for the Helmholtz operator in $\mathbb{H}^N$. As a byproduct, we obtain simple counterexamples to certain Strichartz estimates.