{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:HC62EUXGEYVTNRM2SQYZW4LDCS","short_pith_number":"pith:HC62EUXG","canonical_record":{"source":{"id":"1302.5567","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-22T12:16:55Z","cross_cats_sorted":[],"title_canon_sha256":"50380c79ee84630a84000ff2d7613e61185fbf13a87d54cf2db39f0969652372","abstract_canon_sha256":"a49e0a32a35fb8641fb14a22c50955472dbba28e5bd3a2c9198c65663cc6e54d"},"schema_version":"1.0"},"canonical_sha256":"38bda252e6262b36c59a94319b71631482f791311efe44b280a292921f49f474","source":{"kind":"arxiv","id":"1302.5567","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5567","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5567v1","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5567","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"HC62EUXGEYVT","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HC62EUXGEYVTNRM2","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HC62EUXG","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:HC62EUXGEYVTNRM2SQYZW4LDCS","target":"record","payload":{"canonical_record":{"source":{"id":"1302.5567","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-22T12:16:55Z","cross_cats_sorted":[],"title_canon_sha256":"50380c79ee84630a84000ff2d7613e61185fbf13a87d54cf2db39f0969652372","abstract_canon_sha256":"a49e0a32a35fb8641fb14a22c50955472dbba28e5bd3a2c9198c65663cc6e54d"},"schema_version":"1.0"},"canonical_sha256":"38bda252e6262b36c59a94319b71631482f791311efe44b280a292921f49f474","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:46.383532Z","signature_b64":"B/OhbmCR18Atf1qXTURfjEiHSyjiSwrNJdTyO6KgnHpUcMNRk/mangbM9T3YhiWzYZDWhUedeEfZl053+0TjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38bda252e6262b36c59a94319b71631482f791311efe44b280a292921f49f474","last_reissued_at":"2026-05-18T03:32:46.382973Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:46.382973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.5567","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":"2026-05-18T03:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xqzrSnsWvl0G45EKHbpYcZauzsvJygIHbZ+K2k6ACvU2Dm10hYVLmeYF1gB5BJ9L5wMKmfyjNMvn7NdKXs0iBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T02:22:57.476296Z"},"content_sha256":"ef6480337a2033b23d3db92935c24e1652d2e9339b541e9f41eb3d72ee9f415b","schema_version":"1.0","event_id":"sha256:ef6480337a2033b23d3db92935c24e1652d2e9339b541e9f41eb3d72ee9f415b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:HC62EUXGEYVTNRM2SQYZW4LDCS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Decay properties of the Hardy-Littlewood-Sobolev systems of the Lane-Emden type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Congming Li, Yutian Lei","submitted_at":"2013-02-22T12:16:55Z","abstract_excerpt":"In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\\{{array}{l}\n  (-\\Delta)^k u=v^q,u>0 \\quad in ~R^n,\n  (-\\Delta)^k v=u^p,v>0 \\quad in ~R^n,\n  {array}. $$ and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations $$ \\{{array}{l} u(x)=\\displaystyle\\int_{R^n}\\frac{v^q(y)dy}{|x-y|^{n-\\alpha}},u>0 \\quad in ~R^n, v(x)=\\displaystyle\\int_{R^n}\\frac{u^p(y)dy}{|x-y|^{n-\\alpha}},u>0 \\quad in ~R^n.\n  {array}. $$ Such an integral system is related to the study the extremal functions of the H"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5567","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":"2026-05-18T03:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/OKge+6fYCB/DC5GwifqCR6VDwVRMAa5BaS+qAvQt2+5R4OCy8Mp2YGl87xH2jd+1OZq+kpwgYmUtIeCJWH4Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T02:22:57.476778Z"},"content_sha256":"d6a9e7a3007ecf55d7b2f7bdd84d6ddc07e91aa90aefae58158e44ef6ef19c99","schema_version":"1.0","event_id":"sha256:d6a9e7a3007ecf55d7b2f7bdd84d6ddc07e91aa90aefae58158e44ef6ef19c99"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HC62EUXGEYVTNRM2SQYZW4LDCS/bundle.json","state_url":"https://pith.science/pith/HC62EUXGEYVTNRM2SQYZW4LDCS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HC62EUXGEYVTNRM2SQYZW4LDCS/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":"2026-08-23T02:22:57Z","links":{"resolver":"https://pith.science/pith/HC62EUXGEYVTNRM2SQYZW4LDCS","bundle":"https://pith.science/pith/HC62EUXGEYVTNRM2SQYZW4LDCS/bundle.json","state":"https://pith.science/pith/HC62EUXGEYVTNRM2SQYZW4LDCS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HC62EUXGEYVTNRM2SQYZW4LDCS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HC62EUXGEYVTNRM2SQYZW4LDCS","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":"a49e0a32a35fb8641fb14a22c50955472dbba28e5bd3a2c9198c65663cc6e54d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-22T12:16:55Z","title_canon_sha256":"50380c79ee84630a84000ff2d7613e61185fbf13a87d54cf2db39f0969652372"},"schema_version":"1.0","source":{"id":"1302.5567","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5567","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5567v1","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5567","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"HC62EUXGEYVT","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HC62EUXGEYVTNRM2","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HC62EUXG","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:d6a9e7a3007ecf55d7b2f7bdd84d6ddc07e91aa90aefae58158e44ef6ef19c99","target":"graph","created_at":"2026-05-18T03:32:46Z","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 study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\\{{array}{l}\n  (-\\Delta)^k u=v^q,u>0 \\quad in ~R^n,\n  (-\\Delta)^k v=u^p,v>0 \\quad in ~R^n,\n  {array}. $$ and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations $$ \\{{array}{l} u(x)=\\displaystyle\\int_{R^n}\\frac{v^q(y)dy}{|x-y|^{n-\\alpha}},u>0 \\quad in ~R^n, v(x)=\\displaystyle\\int_{R^n}\\frac{u^p(y)dy}{|x-y|^{n-\\alpha}},u>0 \\quad in ~R^n.\n  {array}. $$ Such an integral system is related to the study the extremal functions of the H","authors_text":"Congming Li, Yutian Lei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-22T12:16:55Z","title":"Decay properties of the Hardy-Littlewood-Sobolev systems of the Lane-Emden type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5567","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:ef6480337a2033b23d3db92935c24e1652d2e9339b541e9f41eb3d72ee9f415b","target":"record","created_at":"2026-05-18T03:32:46Z","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":"a49e0a32a35fb8641fb14a22c50955472dbba28e5bd3a2c9198c65663cc6e54d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-22T12:16:55Z","title_canon_sha256":"50380c79ee84630a84000ff2d7613e61185fbf13a87d54cf2db39f0969652372"},"schema_version":"1.0","source":{"id":"1302.5567","kind":"arxiv","version":1}},"canonical_sha256":"38bda252e6262b36c59a94319b71631482f791311efe44b280a292921f49f474","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38bda252e6262b36c59a94319b71631482f791311efe44b280a292921f49f474","first_computed_at":"2026-05-18T03:32:46.382973Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:46.382973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"B/OhbmCR18Atf1qXTURfjEiHSyjiSwrNJdTyO6KgnHpUcMNRk/mangbM9T3YhiWzYZDWhUedeEfZl053+0TjAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:46.383532Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.5567","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef6480337a2033b23d3db92935c24e1652d2e9339b541e9f41eb3d72ee9f415b","sha256:d6a9e7a3007ecf55d7b2f7bdd84d6ddc07e91aa90aefae58158e44ef6ef19c99"],"state_sha256":"be5dff063137b49b36ea51204206ad9816ac5161b85e98a8305d85a941f9d012"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uxxWpoAMfJ+7vUF39bSJgDt3VH5XzuHs6IofaSfwBa9YPuxCmVU8A2hvttP0uUqKKdTU3uCGJM0CGgZ7YEuxDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T02:22:57.481488Z","bundle_sha256":"717d3d3695c0be47203d3ea1bbcd2c5f590ba0977e78ba81eb15e7e28db02f93"}}