{"paper":{"title":"Radial symmetry of positive solutions involving the fractional Laplacian","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Patricio Felmer, Ying Wang","submitted_at":"2013-11-27T12:33:30Z","abstract_excerpt":"The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-\\Delta)^{\\alpha} u=f(u)+g,\\ \\ {\\rm{in}}\\ \\ B_1, \\quad u=0\\ \\ {\\rm in}\\ \\ B_1^c, where $(-\\Delta)^\\alpha$ denotes the fractional Laplacian, $\\alpha\\in(0,1)$, and $B_1$ denotes the open unit ball centered at the origin in $\\R^N$ with $N\\ge2$. The function $f:[0,\\infty)\\to\\R$ is assumed to be locally Lipschitz continuous and $g: B_1\\to\\R$ is radially symmetric and decreasing in $|x|$.\n  I"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6952","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"}