On Helmholtz Equations and Counterexamples to Strichartz Estimates in Hyperbolic Space

Abstract

Abstract In this paper, we study nonlinear Helmholtz equations (NLH)$$ \begin{equation} \tag{(NLH)} -\Delta_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -\lambda^2 u = \Gamma|u|^{p-2}u \quad\text{in}\ \mathbb{H}^N, \;N\geq 2, \end{equation}$$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 rate of radial solutions is analyzed, with 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.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF
• Repository KITopen (Karlsruhe Institute of Technology) - View - PDF