{"paper":{"title":"A pointwise inequality for a biharmonic equation with negative exponent and related problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qu\\^oc Anh Ng\\^o, Quoc Hung Phan, Van Hoang Nguyen","submitted_at":"2017-05-08T02:46:05Z","abstract_excerpt":"Inspired by a recent pointwise differential inequality for positive bounded solutions of the fourth-order H\\'enon equation $\\Delta^2 u = |x|^a u^p$ in ${\\mathbb R}^n$ with $a \\geqslant 0$, $p > 1$, $n \\geqslant 5$ due to Fazly, Wei, and Xu [ Anal. PDE., 8(2015) 1541--1563], first for some positive constants $\\alpha$ and $\\beta$ we establish the following pointwise inequality\n  \\[\n  \\Delta u \\geqslant \\alpha u^{-\\frac{q-1}2} + \\beta u^{-1} |\\nabla u|^2\n  \\] in ${\\mathbb R}^n$ with $n \\geqslant 3$ for positive $C^4$-solutions of the fourth-order equation\n  \\[\n  \\Delta^2u=-u^{-q} \\quad \\text{ in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02726","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"}