{"paper":{"title":"Existence, regularity, asymptotic decay and radiality of solutions to some extension problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aldo H. S. Medeiros, G. A. Pereira, Hamilton Bueno","submitted_at":"2019-06-20T14:55:41Z","abstract_excerpt":"Supposing only that $\\displaystyle\\lim_{t \\to 0} \\frac{f(t)}{t} = 0$ and $\\displaystyle\\lim_{t \\to \\infty} \\frac{f(t)}{t^{p}} = 0$, for some $p \\in \\left(1,\\frac{N+1}{N-1}\\right)$, we prove that solutions to the extension problem \\begin{equation*}\\left\\{ \\begin{array}{rcll} -\\Delta u+ m^2u &=& 0, &\\mbox{in} \\ \\ \\mathbb{R}^{N+1}_{+} \\\\ -\\frac{\\partial u}{\\partial{x}} (0,y)& =& f(u(0,y)), & y \\in \\mathbb{R}^{N}, \\end{array}\\right. \\end{equation*} and also to the extension Hartree problem \\begin{equation*} \\left\\{\\begin{aligned} -\\Delta u +m^2u&=0, &&\\mbox{in} \\ \\mathbb{R}^{N+1}_+,\\\\ -\\displaysty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.09147","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"}