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This gives $u^\\ast\\in L^\\frac{N}{N-4}$, if N>=5 and $u^*\\in H_0^1$, if N=6."},"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":"1206.6233","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-27T11:30:50Z","cross_cats_sorted":[],"title_canon_sha256":"78dfae65d46c461863c79719f92bab9e188050cfdb3986821b3e40298dc5f45e","abstract_canon_sha256":"7bb5cd63b72435d673358aabe7e1408089268cfc68c15ab57f0bd6d483f39da9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:27.271589Z","signature_b64":"3sS7httjVZZrJMOi6nn/eX3Ae3IetFIP1vX5dRoDJVWs7EzhsQnn5e0pJ9NNk+/Nbqa8QV+I8PU/T16Uv7kLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32a5a744610167bccfa16ef82e66893ebb2e3077a94de36fb4d8d84a0609f858","last_reissued_at":"2026-05-18T03:52:27.270759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:27.270759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundedness of the extremal solutions in dimension 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Salvador Villegas","submitted_at":"2012-06-27T11:30:50Z","abstract_excerpt":"In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation $-\\Delta u=\\lambda f(u)$, in a general smooth bounded domain Omega of R^N, with Dirichlet data $u|_{\\partial \\Omega}=0$, where f is a C^1 positive, nondecreasing and convex function in [0,\\infty) such that $f(s)/s\\rightarrow\\infty$ as $s\\rightarrow\\infty$.\n  In addition, we prove that, for N>=5, the extremal solution $u^*\\in W^{2,\\frac{N}{N-2}}$. 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