{"paper":{"title":"A pointwise inequality for the fourth order Lane-Emden equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juncheng Wei, Mostafa Fazly, Xingwang Xu","submitted_at":"2013-10-08T20:38:14Z","abstract_excerpt":"We prove that the following pointwise inequality holds\n  \\begin{equation*} -\\Delta u \\ge \\sqrt\\frac{2}{(p+1)-c_n} |x|^{\\frac{a}{2}} u^{\\frac{p+1}{2}} + \\frac{2}{n-4} \\frac{|\\nabla u|^2}{u} \\ \\ \\text{in}\\ \\ \\mathbb{R}^n\n  \\end{equation*}\n  where $c_n:=\\frac{8}{n(n-4)}$, for positive bounded solutions of the fourth order H\\'{e}non equation that is \\begin{equation*} \\Delta^2 u = |x|^a u^p \\ \\ \\ \\ \\text {in }\\ \\ \\mathbb{R}^n \\end{equation*} for some $a\\ge0$ and $p>1$. Motivated by the Moser's proof of the Harnack's inequality as well as Moser iteration type arguments in the regularity theory, we d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2275","kind":"arxiv","version":3},"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"}