{"paper":{"title":"$W^{1,q}$ estimates for the extremal solution of reaction-diffusion problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Manel Sanchon","submitted_at":"2012-09-07T13:11:41Z","abstract_excerpt":"We establish a new $W^{1,2\\frac{n-1}{n-2}}$ estimate for the extremal solution of $-\\Delta u=\\lambda f(u)$ in a smooth bounded domain $\\Omega$ of $\\mathbb{R}^n$, which is convex, for arbitrary positive and increasing nonlinearities $f\\in C^1(\\mathbb{R})$ satisfying $\\lim_{t\\rightarrow +\\infty}f(t)/t=+\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1526","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"}