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We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (\\ref{system_abstract}) (resp. (\\ref{fourth_abstract})) provided $ N \\le 10$ and $ 2 \\le p \\le \\theta$ (resp. $ N \\le 10$ and $1 < \\theta"},"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":"1207.1081","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-04T18:55:44Z","cross_cats_sorted":[],"title_canon_sha256":"9dc075fd57efca2efb0b27f35b70f86267e128791640ec0e75e121412e556694","abstract_canon_sha256":"da781660e9133c99339c5e467ccf1a313080bf371104627db15758955ca4bf6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:17.171989Z","signature_b64":"VcXkMeU2wGRZi3NAx++J0p5CNVdlF69H7egOEHZ2TYyph9uhlb4KWHNGx1jkRHiYkGQj0/xlqWeGESgIwSRCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71f685b83bdf46cee7cb9b2530eb01fdb892717f23c279ac04963b44c38ecb3a","last_reissued_at":"2026-05-18T03:18:17.171248Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:17.171248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Liouville theorems for stable Lane-Emden systems and biharmonic problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan","submitted_at":"2012-07-04T18:55:44Z","abstract_excerpt":"We examine the elliptic system given by {equation} \\label{system_abstract}\n  -\\Delta u = v^p, \\qquad -\\Delta v = u^\\theta, \\qquad \\{in} \\IR^N, {equation} for $ 1 < p \\le \\theta$ and the fourth order scalar equation {equation} \\label{fourth_abstract} \\Delta^2 u = u^\\theta, \\qquad \\{in $ \\IR^N$,} {equation} where $ 1 < \\theta$. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (\\ref{system_abstract}) (resp. 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