{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:CYXERF544DAJJZCZEIDXQZNZG6","short_pith_number":"pith:CYXERF54","canonical_record":{"source":{"id":"1805.09150","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-21T12:44:16Z","cross_cats_sorted":[],"title_canon_sha256":"c63d7b1d7dff043b6f87420a55cc2c2f1035cb907fec1980728a0886b4457cfe","abstract_canon_sha256":"dc1c1f06a41e52ecd2c70b96883712aca3ddd7fb9f788eca296466fdf962cbcf"},"schema_version":"1.0"},"canonical_sha256":"162e4897bce0c094e45922077865b937815ecf171f8d045918fb28dd2f2daf27","source":{"kind":"arxiv","id":"1805.09150","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.09150","created_at":"2026-05-18T00:14:53Z"},{"alias_kind":"arxiv_version","alias_value":"1805.09150v2","created_at":"2026-05-18T00:14:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09150","created_at":"2026-05-18T00:14:53Z"},{"alias_kind":"pith_short_12","alias_value":"CYXERF544DAJ","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"CYXERF544DAJJZCZ","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"CYXERF54","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:CYXERF544DAJJZCZEIDXQZNZG6","target":"record","payload":{"canonical_record":{"source":{"id":"1805.09150","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-21T12:44:16Z","cross_cats_sorted":[],"title_canon_sha256":"c63d7b1d7dff043b6f87420a55cc2c2f1035cb907fec1980728a0886b4457cfe","abstract_canon_sha256":"dc1c1f06a41e52ecd2c70b96883712aca3ddd7fb9f788eca296466fdf962cbcf"},"schema_version":"1.0"},"canonical_sha256":"162e4897bce0c094e45922077865b937815ecf171f8d045918fb28dd2f2daf27","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:53.115794Z","signature_b64":"E1PuMlZDX1TMTd13lkZ53oOtbLemmCs4/oTr9A+AQGqHEo1Ag6EEQvF7YRFLcTDhRfmLT53bd8DVEohZMoz6AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"162e4897bce0c094e45922077865b937815ecf171f8d045918fb28dd2f2daf27","last_reissued_at":"2026-05-18T00:14:53.115015Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:53.115015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.09150","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5dMH3zghxXXuSSgOGA6sVLEEKIWXEk58EJkZnrKAfQQ74ehGWEBP4PNateOsoavAeeHlkMFiHt/D6GmKP+R0CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:46:13.903264Z"},"content_sha256":"8851f19f6ee63691082920365e749a25a781ec35e937e5be3f8d13e4478c59c2","schema_version":"1.0","event_id":"sha256:8851f19f6ee63691082920365e749a25a781ec35e937e5be3f8d13e4478c59c2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:CYXERF544DAJJZCZEIDXQZNZG6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finite many critical problems involving Fractional Laplacians in $\\mathbb{R}^{N}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Haibi Chen, Yu Su","submitted_at":"2018-05-21T12:44:16Z","abstract_excerpt":"In this paper, we consider the nonlocal elliptic problems in $\\mathbb{R}^{N}$, which involve finite many critical exponents. By using endpoint refined Hardy--Sobolev inequality, fractional Coulomb--Sobolev space and variational method, we establish the existence of nonnegative solution. Our results generalize some results obtained by Yang and Wu [Adv. Nonlinear Stud. (2017) \\cite{Yang2017}]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09150","kind":"arxiv","version":2},"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":"2026-05-18T00:14:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LlF3TUII9rHiG22IFSGlZuc1jJb3hgthz1e2j4yZxeve/I6EtVyAy+BadfW8n46gzuS/I5FDJk28rXfj+qw4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:46:13.903623Z"},"content_sha256":"5cd4003a9bace0dc533bcf7d1001841560aa155241b24e6d5a64744f50b16fe0","schema_version":"1.0","event_id":"sha256:5cd4003a9bace0dc533bcf7d1001841560aa155241b24e6d5a64744f50b16fe0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CYXERF544DAJJZCZEIDXQZNZG6/bundle.json","state_url":"https://pith.science/pith/CYXERF544DAJJZCZEIDXQZNZG6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CYXERF544DAJJZCZEIDXQZNZG6/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":"2026-06-24T18:46:13Z","links":{"resolver":"https://pith.science/pith/CYXERF544DAJJZCZEIDXQZNZG6","bundle":"https://pith.science/pith/CYXERF544DAJJZCZEIDXQZNZG6/bundle.json","state":"https://pith.science/pith/CYXERF544DAJJZCZEIDXQZNZG6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CYXERF544DAJJZCZEIDXQZNZG6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CYXERF544DAJJZCZEIDXQZNZG6","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":"dc1c1f06a41e52ecd2c70b96883712aca3ddd7fb9f788eca296466fdf962cbcf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-21T12:44:16Z","title_canon_sha256":"c63d7b1d7dff043b6f87420a55cc2c2f1035cb907fec1980728a0886b4457cfe"},"schema_version":"1.0","source":{"id":"1805.09150","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.09150","created_at":"2026-05-18T00:14:53Z"},{"alias_kind":"arxiv_version","alias_value":"1805.09150v2","created_at":"2026-05-18T00:14:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09150","created_at":"2026-05-18T00:14:53Z"},{"alias_kind":"pith_short_12","alias_value":"CYXERF544DAJ","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"CYXERF544DAJJZCZ","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"CYXERF54","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:5cd4003a9bace0dc533bcf7d1001841560aa155241b24e6d5a64744f50b16fe0","target":"graph","created_at":"2026-05-18T00:14:53Z","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 nonlocal elliptic problems in $\\mathbb{R}^{N}$, which involve finite many critical exponents. By using endpoint refined Hardy--Sobolev inequality, fractional Coulomb--Sobolev space and variational method, we establish the existence of nonnegative solution. Our results generalize some results obtained by Yang and Wu [Adv. Nonlinear Stud. (2017) \\cite{Yang2017}].","authors_text":"Haibi Chen, Yu Su","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-21T12:44:16Z","title":"Finite many critical problems involving Fractional Laplacians in $\\mathbb{R}^{N}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09150","kind":"arxiv","version":2},"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":"sha256:8851f19f6ee63691082920365e749a25a781ec35e937e5be3f8d13e4478c59c2","target":"record","created_at":"2026-05-18T00:14:53Z","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":"dc1c1f06a41e52ecd2c70b96883712aca3ddd7fb9f788eca296466fdf962cbcf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-21T12:44:16Z","title_canon_sha256":"c63d7b1d7dff043b6f87420a55cc2c2f1035cb907fec1980728a0886b4457cfe"},"schema_version":"1.0","source":{"id":"1805.09150","kind":"arxiv","version":2}},"canonical_sha256":"162e4897bce0c094e45922077865b937815ecf171f8d045918fb28dd2f2daf27","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"162e4897bce0c094e45922077865b937815ecf171f8d045918fb28dd2f2daf27","first_computed_at":"2026-05-18T00:14:53.115015Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:53.115015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E1PuMlZDX1TMTd13lkZ53oOtbLemmCs4/oTr9A+AQGqHEo1Ag6EEQvF7YRFLcTDhRfmLT53bd8DVEohZMoz6AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:53.115794Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.09150","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8851f19f6ee63691082920365e749a25a781ec35e937e5be3f8d13e4478c59c2","sha256:5cd4003a9bace0dc533bcf7d1001841560aa155241b24e6d5a64744f50b16fe0"],"state_sha256":"ca8e6aa8e1a3171b01f2abe187395060198787e93b89291f53b86b6594c3acba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Tx2zZXqUKzNDHSTY8G4Yj+tQ3mW9t1vNeKCe8YDvRThJqgkP7ZL0Cr1R+nM2BaLfS1cbcO5IwmQNJSz2k3wBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T18:46:13.905644Z","bundle_sha256":"6554ae711de4a7dde19bb11b2d234201b96002baac19601a3301987f714124d5"}}