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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1906.03739","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-10T00:06:52Z","cross_cats_sorted":[],"title_canon_sha256":"6d8f42cf155b43532a8a9080da458ac098ea8b6193ad1ba90ca27a3ece229c74","abstract_canon_sha256":"9a17bb44535d60ec7d59693d71315fbb3b5c900c1a3a16b9c9dfc542eeb35355"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:45.102875Z","signature_b64":"FHmYci4Hp6XdJHq0r7a5a+YjLLjV73EL1hhS/O1UyXrl7e1tY0sFYDUvf6tu1nvlHMHy4i+u97MagVV8BXNeDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a988fca5d7715f3aa9db843007fdb257fc07ae9e6c3ff327bbdd9328b7005ed","last_reissued_at":"2026-05-17T23:43:45.102227Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:45.102227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $a, \\ b>0$. 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