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Mishra","submitted_at":"2018-07-30T06:53:17Z","abstract_excerpt":"In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\\\ \\begin{equation} \\left\\{ \\begin{array}{rllll} \\mc L_M(u)&=\\lambda f(x)|u|^{q-2}u+ \\frac{2\\alpha}{\\alpha+\\beta}\\left|u\\right|^{\\alpha-2}u|v|^\\beta & \\text{in } \\Omega,\\\\ \\mc L_M(v)&=\\mu g(x)|v|^{q-2}v+ \\frac{2\\beta}{\\alpha+\\beta}\\left|u\\right|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega,\\\\ u&=v=0 &\\mbox{in } \\mathbb{R}^{N}\\setminus \\Omega, \\end{array} \\right. \\end{equation} where $\\mc L_M(u)=M\\left(\\displaystyle \\int_\\Omega|(-\\Delta)^{\\frac{s}{2}}u|^2dx\\right)(-\\Delta)^{s} u"},"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":"1807.11191","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-30T06:53:17Z","cross_cats_sorted":[],"title_canon_sha256":"663eeec34afb27049041c337596ffbaa5ab6cc83a44508eddc915840f122e153","abstract_canon_sha256":"4e2c8c187df5c6846f602ef5f45a8d7dcc723431fea6cf594f81894b60f4abbb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:32.691845Z","signature_b64":"w4Lo25QkP12W36vwW3PbPNMta94nanb+mHvhmBUnIRBw/ftfydSzku4+XZl3X86j1G62Mh1vTNkspnwvMXwEAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed0b04447a3cca6bef644ef01d7d99b85ab25272aabc2e2a95bdaaa6974103d1","last_reissued_at":"2026-05-18T00:09:32.691465Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:32.691465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nehari Manifold for fractional Kirchhoff system with critical nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"J. Giacomoni, J.M. do \\'O, P.K. Mishra","submitted_at":"2018-07-30T06:53:17Z","abstract_excerpt":"In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\\\ \\begin{equation} \\left\\{ \\begin{array}{rllll} \\mc L_M(u)&=\\lambda f(x)|u|^{q-2}u+ \\frac{2\\alpha}{\\alpha+\\beta}\\left|u\\right|^{\\alpha-2}u|v|^\\beta & \\text{in } \\Omega,\\\\ \\mc L_M(v)&=\\mu g(x)|v|^{q-2}v+ \\frac{2\\beta}{\\alpha+\\beta}\\left|u\\right|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega,\\\\ u&=v=0 &\\mbox{in } \\mathbb{R}^{N}\\setminus \\Omega, \\end{array} \\right. \\end{equation} where $\\mc L_M(u)=M\\left(\\displaystyle \\int_\\Omega|(-\\Delta)^{\\frac{s}{2}}u|^2dx\\right)(-\\Delta)^{s} u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.11191","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1807.11191","created_at":"2026-05-18T00:09:32.691529+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.11191v1","created_at":"2026-05-18T00:09:32.691529+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.11191","created_at":"2026-05-18T00:09:32.691529+00:00"},{"alias_kind":"pith_short_12","alias_value":"5UFQIRD2HTFG","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"5UFQIRD2HTFGX33E","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"5UFQIRD2","created_at":"2026-05-18T12:32:08.215937+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB","json":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB.json","graph_json":"https://pith.science/api/pith-number/5UFQIRD2HTFGX33EJ3YB27MZXB/graph.json","events_json":"https://pith.science/api/pith-number/5UFQIRD2HTFGX33EJ3YB27MZXB/events.json","paper":"https://pith.science/paper/5UFQIRD2"},"agent_actions":{"view_html":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB","download_json":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB.json","view_paper":"https://pith.science/paper/5UFQIRD2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.11191&json=true","fetch_graph":"https://pith.science/api/pith-number/5UFQIRD2HTFGX33EJ3YB27MZXB/graph.json","fetch_events":"https://pith.science/api/pith-number/5UFQIRD2HTFGX33EJ3YB27MZXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB/action/storage_attestation","attest_author":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB/action/author_attestation","sign_citation":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB/action/citation_signature","submit_replication":"https://pith.science/pith/5UFQIRD2HTFGX33EJ3YB27MZXB/action/replication_record"}},"created_at":"2026-05-18T00:09:32.691529+00:00","updated_at":"2026-05-18T00:09:32.691529+00:00"}