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We obtain the radial symmetry and monotonicity properties for nonnegative viscosity solutions of begin{equation}F (D^2 u)+u^p=0 \\quad \\quad \\mbox{in}\\ \\mathbb R^n \\label{abs}\\end{equation} under the asymptotic decay rate $u=o(|x|^{-\\frac{2}{p-1}})$ at infinity, where $p>1$ (Theorem 1, Corollary 1). As a consequence of our symmetry results, we obtain the nonexistence of any nontrivial and nonnegative solution when $F$ is the Pucci extremal operators (Co"},"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":"1301.7141","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-01-30T04:40:45Z","cross_cats_sorted":[],"title_canon_sha256":"6151f35d973e9b5726ec5d9f38538db69bc7fcf467609c4d98f8f27b97a5dbcb","abstract_canon_sha256":"34731ac8bb4cd779412161346a813a009980cefcd5268d4fca9b284d5bf0f4f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:01.202820Z","signature_b64":"UwsxdpmofvvKa8DyDYf6Jh0ilImNR2w67qEEd9wZ+cFcW2SZUYj+pPZ/zW9G80X7rQZ/6mVpv9+58T5JfPwaCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd1214520131caaa9ae96beb6c301d4f4e5ebf1af1b17c6254c7d0db522c7823","last_reissued_at":"2026-05-18T03:35:01.201835Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:01.201835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guozhen Lu, Jiuyi Zhu","submitted_at":"2013-01-30T04:40:45Z","abstract_excerpt":"This paper is concerned about maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. 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