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We show the regularity of all semi-stable solutions and hence of the extremal solutions, provided [N < 2 + 4 \\sqrt{2} + 4 \\sqrt{2 - \\sqrt{2}} \\approx 10.718 when $ f(u)=e^u$,] and [\\frac{N}{4} < \\frac{p}{p-1} + \\frac{p+1}{p-1} (\\sqrt{\\frac{2p}{p+1}} + \\sqrt{\\frac{2p}{p"},"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":"1206.3471","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-15T14:03:32Z","cross_cats_sorted":[],"title_canon_sha256":"c65bcb3d46370802752f1c3710e83135f88cc02fa6b12248684ec74ce763b78d","abstract_canon_sha256":"965c6d4fe9266c9cb1995ef059e5e8453c30df85150c8d4f3b6fbc2d32012738"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:30.173713Z","signature_b64":"9CGCZ0UrakPvLe/PDaAiPRSOInnEMuJ7b3RKb1bKwwpHJwe4sazU91WU5NBii9PHgFMjSCBfrTX6NwGhV5EADw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc882ae5c7361025b0bf8133d61d3bd2c125d8250f7bdbee4026a73a6eee5096","last_reissued_at":"2026-05-18T03:53:30.172929Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:30.172929Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan, Nassif Ghoussoub","submitted_at":"2012-06-15T14:03:32Z","abstract_excerpt":"We examine the fourth order problem $\\Delta^2 u = \\lambda f(u) $ in $ \\Omega$ with $ \\Delta u = u =0 $ on $ \\partial \\Omega$, where $ \\lambda > 0$ is a parameter, $ \\Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$, $ f(u)=(1+u)^p $ or $ f(u)= \\frac{1}{(1-u)^p}$ where $ p>1$. 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