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On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space

Casteras, Jean-Baptiste; Mandel, Rainer

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.

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DOI: 10.5445/IR/1000091019
Veröffentlicht am 14.02.2019
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2019
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000091019
Verlag KIT, Karlsruhe
Umfang 17 S.
Serie CRC 1173 ; 2019/5
Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
Schlagwörter Helmholtz equations, hyperbolic space, resolvent estimates, Strichartz estimates
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft
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