{"paper":{"title":"On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean-Baptiste Casteras, Rainer Mandel","submitted_at":"2019-02-12T12:14:23Z","abstract_excerpt":"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 spac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.04351","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}