{"paper":{"title":"Nonlinear elliptic equations on the upper half space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lei Wang, Meijun Zhu, Sufanf Tang","submitted_at":"2019-06-10T00:06:52Z","abstract_excerpt":"In this paper we shall classify all positive solutions of $ \\Delta u =a u^p$ on the upper half space $ H =\\Bbb{R}_+^n$ with nonlinear boundary condition $ {\\partial u}/{\\partial t}= - b u^q $ on $\\partial H$ for both positive parameters $a, \\ b>0$. We will prove that for $p \\ge {(n+2)}/{(n-2)}, 1\\leq q<{n}/{(n-2)}$ (and $n \\ge 3$) all positive solutions are functions of last variable; for $p= {(n+2)}/{(n-2)}, q= {n}/{(n-2)}$ (and $n \\ge 3$) positive solutions must be either some functions depending only on last variable, or radially symmetric functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.03739","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"}