{"paper":{"title":"Biharmonic equation with singular nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"G. Warnault, J. Giacomoni, S. Prashanth","submitted_at":"2015-11-12T16:26:14Z","abstract_excerpt":"We consider the following problem: \\begin{eqnarray*} ( P)\\qquad \\displaystyle\\left\\{\\begin{array} {ll}\n  & \\Delta^2 u\n  = K(x)u^{-\\alpha}\n  \\quad \\mbox{ in }\\,\\Omega , \\\\ &u> 0\\quad \\mbox{ in }\\,\\Omega, \\;\\;u\\vert_{\\partial\\Omega}=0, \\,\\Delta u\\vert_{\\partial\\Omega} = 0. \\end{array}\\right.\n  \\end{eqnarray*} We prove the main existence result:\n  Assume that $\\alpha+\\beta<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \\begin{eqnarray}\\label{behaviour-bound} c_1 \\rho(x)\\leq u(x)\\leq c_2 \\rho(x) \\end{eqnarray}\n  where $\\rho(x)=d(x,\\partial\\Ome"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03948","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"}