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He","submitted_at":"2016-07-05T11:49:23Z","abstract_excerpt":"We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \\begin{equation*} M\\left(\\int_\\Omega|(-\\Delta)^{\\frac{\\alpha}{2}}u|^2dx\\right)(-\\Delta)^{\\alpha} u= \\lambda f(x)|u|^{q-2}u+|u|^{2^*_\\alpha-2}u\\;\\; \\text{in}\\; \\Omega,\\;u=0\\;\\textrm{in}\\;\\mathbb R^n\\setminus \\Omega, \\end{equation*} where $\\Omega\\subset \\mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\\varepsilon t, \\; a, \\; \\varepsilon>0,\\; 0<\\alpha<1, \\; 2\\alpha<n<4\\alpha$ and $ \\; 1<q<2$. Here $2^*_\\alpha={2n}/{(n-2\\alpha)}$ is the fractional cri"},"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":"1607.01200","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-05T11:49:23Z","cross_cats_sorted":[],"title_canon_sha256":"a8413511e1345e44e8b6d50937113d5dfcba717659fc9bc91362c50159339805","abstract_canon_sha256":"00815080e0de868983d4bc2fd6af688b61a59dffadf7cc8b6a6eefe93f72ad1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:36.670659Z","signature_b64":"hlMUn1/ogZU8J9Nh566q297zuuG5eCzN0+vZT9h2oxaUyahbCIceRvFFt5iHUMfHXkoEAw6Tu79NQTzIw0ReDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49dfc2b7b61e05e840ac1e14f380395e346ed78cf38bc8f6508a0ba1094a4f9a","last_reissued_at":"2026-05-18T00:27:36.669748Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:36.669748Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractional Kirchhoff problem with critical indefinite nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"J. M. do \\'O, P. K. Mishra, X. He","submitted_at":"2016-07-05T11:49:23Z","abstract_excerpt":"We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \\begin{equation*} M\\left(\\int_\\Omega|(-\\Delta)^{\\frac{\\alpha}{2}}u|^2dx\\right)(-\\Delta)^{\\alpha} u= \\lambda f(x)|u|^{q-2}u+|u|^{2^*_\\alpha-2}u\\;\\; \\text{in}\\; \\Omega,\\;u=0\\;\\textrm{in}\\;\\mathbb R^n\\setminus \\Omega, \\end{equation*} where $\\Omega\\subset \\mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\\varepsilon t, \\; a, \\; \\varepsilon>0,\\; 0<\\alpha<1, \\; 2\\alpha<n<4\\alpha$ and $ \\; 1<q<2$. 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