{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MJWPRN4XVTD7FQCD6UVYBOHAM6","short_pith_number":"pith:MJWPRN4X","canonical_record":{"source":{"id":"1605.06906","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-23T06:41:57Z","cross_cats_sorted":[],"title_canon_sha256":"d7dfe8a7c04182fced1a9b7a019bb256a6d094c07448b7c046b2449e28f48077","abstract_canon_sha256":"1158306fee673bf0b1b25063f879fc0ab8f96646b1ffa83012641a5e7c91011b"},"schema_version":"1.0"},"canonical_sha256":"626cf8b797acc7f2c043f52b80b8e067878dee43bd6de85f687d2334d43036c7","source":{"kind":"arxiv","id":"1605.06906","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06906","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06906v1","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06906","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"pith_short_12","alias_value":"MJWPRN4XVTD7","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MJWPRN4XVTD7FQCD","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MJWPRN4X","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MJWPRN4XVTD7FQCD6UVYBOHAM6","target":"record","payload":{"canonical_record":{"source":{"id":"1605.06906","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-23T06:41:57Z","cross_cats_sorted":[],"title_canon_sha256":"d7dfe8a7c04182fced1a9b7a019bb256a6d094c07448b7c046b2449e28f48077","abstract_canon_sha256":"1158306fee673bf0b1b25063f879fc0ab8f96646b1ffa83012641a5e7c91011b"},"schema_version":"1.0"},"canonical_sha256":"626cf8b797acc7f2c043f52b80b8e067878dee43bd6de85f687d2334d43036c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:10.905666Z","signature_b64":"X8QIjwI0cW3TptQCuVAfUPJudENtjpz3k+6qmAzMI26qI6kkzB/Ei9drRw/oQn0FivbAvbeSQB9Iui+vVcR5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"626cf8b797acc7f2c043f52b80b8e067878dee43bd6de85f687d2334d43036c7","last_reissued_at":"2026-05-18T01:14:10.904937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:10.904937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.06906","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:14:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CxDSJrJtbfl4Q39ud5enSXUy9HU4dlvdB+CH4UQ/oyvUWCJadNlB9U6MsLm7cYHmoa+hQ33GW/Gm8xLRcPSHAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T17:53:36.880074Z"},"content_sha256":"a7b02914f45e2eaf47ac4fa10dd70ab26e676bc036f42c119251ac872028a4f2","schema_version":"1.0","event_id":"sha256:a7b02914f45e2eaf47ac4fa10dd70ab26e676bc036f42c119251ac872028a4f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MJWPRN4XVTD7FQCD6UVYBOHAM6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a critical Kirchhoff problem in high dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yisheng Huang, Yuanze Wu, Zeng Liu","submitted_at":"2016-05-23T06:41:57Z","abstract_excerpt":"In this paper, we consider the following Kirchhoff problem $$ \\left\\{\\aligned -\\bigg(a+b\\int_{\\Omega}|\\nabla u|^2dx\\bigg)\\Delta u&= \\lambda u^{q-1} + \\mu u^{2^*-1}, &\\quad \\text{in }\\Omega, \\\\ u&>0,&\\quad\\text{in }\\Omega,\\\\ u&=0,&\\quad\\text{on }\\partial\\Omega, \\endaligned \\right.\\eqno{(\\mathcal{P})} $$ where $\\Omega\\subset \\bbr^N(N\\geq4)$ is a bounded domain, $2\\leq q<2^*$, $2^*=\\frac{2N}{N-2}$ is the critical Sobolev exponent and $a$, $b$, $\\lambda$, $\\mu$ are positive parameters. By using the variational method, we obtain some existence and nonexistence results to $(\\mathcal{P})$ for all $N\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06906","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:14:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XHYtzRxQCnxNIB9HHrvP95BCQU5TfpnQIUg1Owemq2rIji8lRvLB9N5WqZyD8BcPgv+JLj4ZKCZAVc06an7JDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T17:53:36.880805Z"},"content_sha256":"e591185a3c1b373166af7c5be799a6426377ac134d1c1ce1509181c14928e55d","schema_version":"1.0","event_id":"sha256:e591185a3c1b373166af7c5be799a6426377ac134d1c1ce1509181c14928e55d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MJWPRN4XVTD7FQCD6UVYBOHAM6/bundle.json","state_url":"https://pith.science/pith/MJWPRN4XVTD7FQCD6UVYBOHAM6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MJWPRN4XVTD7FQCD6UVYBOHAM6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T17:53:36Z","links":{"resolver":"https://pith.science/pith/MJWPRN4XVTD7FQCD6UVYBOHAM6","bundle":"https://pith.science/pith/MJWPRN4XVTD7FQCD6UVYBOHAM6/bundle.json","state":"https://pith.science/pith/MJWPRN4XVTD7FQCD6UVYBOHAM6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MJWPRN4XVTD7FQCD6UVYBOHAM6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MJWPRN4XVTD7FQCD6UVYBOHAM6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1158306fee673bf0b1b25063f879fc0ab8f96646b1ffa83012641a5e7c91011b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-23T06:41:57Z","title_canon_sha256":"d7dfe8a7c04182fced1a9b7a019bb256a6d094c07448b7c046b2449e28f48077"},"schema_version":"1.0","source":{"id":"1605.06906","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06906","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06906v1","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06906","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"pith_short_12","alias_value":"MJWPRN4XVTD7","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MJWPRN4XVTD7FQCD","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MJWPRN4X","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:e591185a3c1b373166af7c5be799a6426377ac134d1c1ce1509181c14928e55d","target":"graph","created_at":"2026-05-18T01:14:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we consider the following Kirchhoff problem $$ \\left\\{\\aligned -\\bigg(a+b\\int_{\\Omega}|\\nabla u|^2dx\\bigg)\\Delta u&= \\lambda u^{q-1} + \\mu u^{2^*-1}, &\\quad \\text{in }\\Omega, \\\\ u&>0,&\\quad\\text{in }\\Omega,\\\\ u&=0,&\\quad\\text{on }\\partial\\Omega, \\endaligned \\right.\\eqno{(\\mathcal{P})} $$ where $\\Omega\\subset \\bbr^N(N\\geq4)$ is a bounded domain, $2\\leq q<2^*$, $2^*=\\frac{2N}{N-2}$ is the critical Sobolev exponent and $a$, $b$, $\\lambda$, $\\mu$ are positive parameters. By using the variational method, we obtain some existence and nonexistence results to $(\\mathcal{P})$ for all $N\\","authors_text":"Yisheng Huang, Yuanze Wu, Zeng Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-23T06:41:57Z","title":"On a critical Kirchhoff problem in high dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06906","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a7b02914f45e2eaf47ac4fa10dd70ab26e676bc036f42c119251ac872028a4f2","target":"record","created_at":"2026-05-18T01:14:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1158306fee673bf0b1b25063f879fc0ab8f96646b1ffa83012641a5e7c91011b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-23T06:41:57Z","title_canon_sha256":"d7dfe8a7c04182fced1a9b7a019bb256a6d094c07448b7c046b2449e28f48077"},"schema_version":"1.0","source":{"id":"1605.06906","kind":"arxiv","version":1}},"canonical_sha256":"626cf8b797acc7f2c043f52b80b8e067878dee43bd6de85f687d2334d43036c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"626cf8b797acc7f2c043f52b80b8e067878dee43bd6de85f687d2334d43036c7","first_computed_at":"2026-05-18T01:14:10.904937Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:10.904937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X8QIjwI0cW3TptQCuVAfUPJudENtjpz3k+6qmAzMI26qI6kkzB/Ei9drRw/oQn0FivbAvbeSQB9Iui+vVcR5Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:10.905666Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.06906","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a7b02914f45e2eaf47ac4fa10dd70ab26e676bc036f42c119251ac872028a4f2","sha256:e591185a3c1b373166af7c5be799a6426377ac134d1c1ce1509181c14928e55d"],"state_sha256":"11337a0f52bba652501975b06e830cbf56f490bce3785fc73da7896279ec3d38"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XUm6dc+oz6CdeKs/FaCFU4DHtiGQNYL0CUvAgKVt0oiuc0VdSVEQ2FRvcglYGvEGnjw0R1s12xP12Prm2GbCDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T17:53:36.884771Z","bundle_sha256":"4353a7996566b60e6353148cb08a24216adbd04274289e652c5857d7130fd59a"}}