{"paper":{"title":"A Liouville theorem for an integral equation of the Ginzburg-Landau type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xin Xu, Yutian Lei","submitted_at":"2020-06-24T22:38:41Z","abstract_excerpt":"In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \\begin{equation*} u(x)=\\overrightarrow{l}+C_*\\int_{\\mathbb{R}^{n}}\\frac{u(1-|u|^{2})}{|x-y|^{n-\\alpha}}dy. \\end{equation*} Here $u: \\mathbb{R}^{n} \\to \\mathbb{R}^{k}$ is a bounded, uniformly continuous and differentiable function with $k \\geq 1$ and $1<\\alpha<n$, $\\overrightarrow{l} \\in \\mathbb{R}^{k}$ is a constant vector, and $C_*$ is a real constant. If $u$ is the finite energy solution, we prove that $|\\overrightarrow{l}| \\in \\{0,1\\}$. Furthermore, we also give a Liouville type theorem (i.e., $u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.14951","kind":"arxiv","version":2},"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/2006.14951/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"}