{"paper":{"title":"Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tom\\'as Sanz-Perela","submitted_at":"2017-12-22T17:55:36Z","abstract_excerpt":"We study the regularity of stable solutions to the problem $$ \\left\\{ \\begin{array}{rcll} (-\\Delta)^s u &=& f(u) & \\text{in} \\quad B_1\\,, u &\\equiv&0 & \\text{in} \\quad \\mathbb R^n\\setminus B_1\\,, \\end{array} \\right. $$ where $s\\in(0,1)$. Our main result establishes an $L^\\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \\leq n < 2(s+2+\\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\\in(0,1)$ in dimensions $2 \\leq n\\leq 6$. It applies to all nonlinearities $f\\in C^2$.\n  For such parameters $s$ and $n$, our result leads to the regular"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08598","kind":"arxiv","version":2},"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"}