{"paper":{"title":"Elliptic equations with nonlinear absorption depending on the solution and its gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Moshe Marcus, Phuoc-Tai Nguyen","submitted_at":"2014-09-25T09:13:34Z","abstract_excerpt":"We study positive solutions of equation (E1) $-\\Delta u + u^p|\\nabla u|^q= 0$ ($0\\leq p$, $0\\leq q\\leq 2$, $p+q>1$) and (E2) $-\\Delta u + u^p + |\\nabla u|^q =0$ ($p>1$, $1<q\\leq 2$) in a smooth bounded domain $\\Omega \\subset \\mathbb{R}^N$. We obtain a sharp condition on $p$ and $q$ under which, for every positive, finite Borel measure $\\mu$ on $\\partial \\Omega$, there exists a solution such that $u=\\mu$ on $\\partial \\Omega$. Furthermore, if the condition mentioned above fails then any isolated point singularity on $\\partial \\Omega$ is removable, namely there is no positive solution that vanish"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7191","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"}