{"paper":{"title":"Properties of the extremal solution for a fourth-order elliptic problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Baishun Lai, Zhuoran Du","submitted_at":"2010-09-14T00:49:31Z","abstract_excerpt":"Let $\\lambda^{*}>0$ denote the largest possible value of $\\lambda$ such that $$ \\{{array}{lllllll} \\Delta^{2}u=\\frac{\\lambda}{(1-u)^{p}} & \\{in}\\ \\ B, 0<u\\leq 1 & \\{in}\\ \\ B, u=\\frac{\\partial u}{\\partial n} =0 & \\{on}\\ \\ \\partial B. {array} . $$ has a solution, where $B$ is the unit ball in $R^{n}$ centered at the origin, $p>1$ and $n$ is the exterior unit normal vector. We show that for $\\lambda=\\lambda^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\\geq 13$ for $p$ large enough and $1-C_{0}r^{\\frac{4}{p+1}}\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2546","kind":"arxiv","version":3},"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"}