{"paper":{"title":"Existence and asymptotics of nonlinear Helmholtz eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrew Hassell, Jacob Shapiro, Jesse Gell-Redman, Junyong Zhang","submitted_at":"2019-08-13T23:53:53Z","abstract_excerpt":"We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \\begin{equation*} (\\Delta - \\lambda^2) u = N[u], \\end{equation*} where $\\Delta = -\\sum_j \\partial^2_j$ is the Laplacian on $\\mathbb{R}^n$ with sign convention that it is positive as an operator, $\\lambda$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\\geq p$ in $u$, $\\overline{u}$ and their derivatives of order up to two, for some $p \\geq 2$. Nonlinear Helmholtz eigenfunctions with $N[u]= \\pm |u|^{p-1} u$ were first con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04890","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.04890/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}