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Then, we apply these formulae to classify stable solutions of the above equation."},"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":"1410.6786","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-24T19:06:06Z","cross_cats_sorted":[],"title_canon_sha256":"886439e68441a347c4f729fcd0c3b101c68cbc979feae2ecd3261585da00a765","abstract_canon_sha256":"a50b66428692435eb722c3d3e39999c1b584c81916ff5572bff2536b589b0f64"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:05.562063Z","signature_b64":"Y8jSp1h0Eeb6hMfDEzBYLHj8G8EqfdW6MzbNrHQVBsGmWbgDxVXH7Xl6LcWeEE56ZiHGCDoouMou3HsqVA58DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae95c8309f35320fb4dcc575ed9b7fb966b71c0fb4f3cb0c8b20bbe4ed7faebf","last_reissued_at":"2026-05-18T01:27:05.561201Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:05.561201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On stable solutions of the fractional Henon-Lane-Emden equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juncheng Wei, Mostafa Fazly","submitted_at":"2014-10-24T19:06:06Z","abstract_excerpt":"We derive monotonicity formulae for solutions of the fractional H\\'{e}non-Lane-Emden equation \\begin{equation*} (-\\Delta)^{s} u=|x|^a |u|^{p-1} u \\ \\ \\ \\text{in } \\ \\ \\mathbb{R}^n, \\end{equation*} when $0<s<2$, $a>0$ and $p>1$. 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