{"paper":{"title":"Classification of entire solutions of $(-\\Delta)^N u + u^{-(4N-1)}= 0$ with exact linear growth at infinity in $\\mathbf R^{2N-1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qu\\^oc-Anh Ng\\^o","submitted_at":"2016-11-07T03:11:02Z","abstract_excerpt":"In this paper, we study global positive $C^{2N}$-solutions of the geometrically interesting equation $(-\\Delta)^N u + u^{-(4N-1)}= 0$ in $\\mathbf R^{2N-1}$. We prove that any $C^{2N}$-solution $u$ of the equation having linear growth at infinity must satisfy the integral equation \\[ u(x) = c_0 \\int_{\\mathbf R^{2N-1}} {|x - y|{u^{-(4N-1)}}(y)dy} \\] for some positive constant $c_0$ and hence takes the following form \\[ u(x) = (1+|x|^2)^{1/2} \\] in $\\mathbf R^{2N-1}$ up to dilations and translations. We also provide several non-existence results for positive $C^{2N}$-solutions of $(-\\Delta)^N u ="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01882","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":""},"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"}