{"paper":{"title":"Liouville theorems for the polyharmonic Henon-Lane-Emden system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly","submitted_at":"2013-08-01T00:54:08Z","abstract_excerpt":"We study Liouville theorems for the following polyharmonic H\\'{e}non-Lane-Emden system \\begin{eqnarray*}\n  \\left\\{\\begin{array}{lcl} (-\\Delta)^m u&=& |x|^{a}v^p \\ \\ \\text{in}\\ \\ \\mathbb{R}^n,\\\\ (-\\Delta)^m v&=& |x|^{b}u^q \\ \\ \\text{in}\\ \\ \\mathbb{R}^n, \\end{array}\\right.\n  \\end{eqnarray*} when $m,p,q \\ge 1,$ $pq\\neq1$, $a,b\\ge0$. The main conjecture states that $(u,v)=(0,0)$ is the unique nonnegative solution of this system whenever $(p,q)$ is {\\it under} the critical Sobolev hyperbola, i.e. $ \\frac{n+a}{p+1}+\\frac{n+b}{q+1}>{n-2m}$. We show that this is indeed the case in dimension $n=2m+1$ f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0073","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"}