{"paper":{"title":"Regularity of the extremal solution for singular p-Laplace equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniele Castorina","submitted_at":"2014-07-14T11:01:03Z","abstract_excerpt":"We study the regularity of the extremal solution $u^*$ to the singular reaction-diffusion problem $-\\Delta_p u = \\lambda f(u)$ in $\\Omega$, $u =0$ on $\\partial \\Omega$, where $1<p<2$, $0 < \\lambda < \\lambda^*$, $\\Omega \\subset \\mathbb{R}^n$ is a smooth bounded domain and $f$ is any positive, superlinear, increasing and (asymptotically) convex $C^1$ nonlinearity. We provide a simple proof of known $L^r$ and $W^{1,r}$ \\textit{a priori} estimates for $u^*$, i.e. $u^* \\in L^\\infty(\\Omega)$ if $n \\leq p+2$, $u^* \\in L^{\\frac{2n}{n-p-2}}(\\Omega)$ if $n > p+2$ and $|\\nabla u^*|^{p-1} \\in L^{\\frac{n}{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3602","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"}