{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:7R6YTMXW2CDGAAL3ELIETBZNDY","short_pith_number":"pith:7R6YTMXW","canonical_record":{"source":{"id":"1308.3115","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-14T13:23:33Z","cross_cats_sorted":[],"title_canon_sha256":"156304b26d0e0c545d7cbedd977ba621a5053c3529b156f425163dc8ad56923f","abstract_canon_sha256":"3c35287c4854217b9ac3f98b164654abcb16fed55800a3d18d6327ad45d5e5d2"},"schema_version":"1.0"},"canonical_sha256":"fc7d89b2f6d08660017b22d049872d1e3b1e388d318b9d43b8c332b91f0cd875","source":{"kind":"arxiv","id":"1308.3115","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.3115","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"arxiv_version","alias_value":"1308.3115v2","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.3115","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"pith_short_12","alias_value":"7R6YTMXW2CDG","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7R6YTMXW2CDGAAL3","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7R6YTMXW","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:7R6YTMXW2CDGAAL3ELIETBZNDY","target":"record","payload":{"canonical_record":{"source":{"id":"1308.3115","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-14T13:23:33Z","cross_cats_sorted":[],"title_canon_sha256":"156304b26d0e0c545d7cbedd977ba621a5053c3529b156f425163dc8ad56923f","abstract_canon_sha256":"3c35287c4854217b9ac3f98b164654abcb16fed55800a3d18d6327ad45d5e5d2"},"schema_version":"1.0"},"canonical_sha256":"fc7d89b2f6d08660017b22d049872d1e3b1e388d318b9d43b8c332b91f0cd875","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:55.348517Z","signature_b64":"H4DTOSIMIrmdS/0o7vTrj4kTEOZinGhVM9zWwLJnAAGLqYobGX4jvevBtKInCs19/PQ7dbdzMusyY0MAlF4dAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc7d89b2f6d08660017b22d049872d1e3b1e388d318b9d43b8c332b91f0cd875","last_reissued_at":"2026-05-18T03:15:55.347731Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:55.347731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.3115","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":"2026-05-18T03:15:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9AOrYZFlgIyv9Xs0Jwnz9szPW28FzVHnY1v2ktLLpu4rh8tmbDVIlhDlxey0/KEx6+9y2yf6pWrU3cLrBcFBDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T07:54:13.384749Z"},"content_sha256":"ecbea8aa3987ba15ed509b13be6b4c160b7230bd62ac4daaaa11112b5b2e0d3f","schema_version":"1.0","event_id":"sha256:ecbea8aa3987ba15ed509b13be6b4c160b7230bd62ac4daaaa11112b5b2e0d3f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:7R6YTMXW2CDGAAL3ELIETBZNDY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positive radial solutions for coupled Schr\\\"{o}dinger system with critical exponent in $\\R^N\\,(N\\geq5)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hong-yu Ye, Yan-fang Peng","submitted_at":"2013-08-14T13:23:33Z","abstract_excerpt":"We study the following coupled Schr\\\"odinger system \\ds -\\Delta u+u=u^{2^*-1}+\\be u^{\\frac{2^*}{2}-1}v^{\\frac{2^*}{2}}+\\la_1u^{\\al-1}, &x\\in \\R^N, \\ds -\\Delta v+v=v^{2^*-1}+\\be u^{\\frac{2^*}{2}}v^{\\frac{2^*}{2}-1}+\\la_2v^{r-1}, &x\\in \\R^N, u,v > 0, &x\\in \\R^N, where $N\\geq 5, \\la_1,\\la_2>0,\\be\\neq 0, 2<\\al,r<2^*,2^*\\triangleq \\frac{2N}{N-2}.$ Note that the nonlinearity and the coupling terms are both critical. Using the Mountain Pass Theorem, Ekeland's variational principle and Nehari mainfold, we show that this critical system has a positive radial solution for positive $\\be$ and some negativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3115","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":"2026-05-18T03:15:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jhdH4Io6LQm3FJtLQOHvYJ6a1r4LxCeHdEskqyKsE2YgC3SRBj29fpg7dD4lDj7TFSNMeMUW7AlSJSbuYKgeCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T07:54:13.385091Z"},"content_sha256":"150ab2d6ef66c0062d882cee901afdb078cec274aaf632f8eae5ba5c44d8cf76","schema_version":"1.0","event_id":"sha256:150ab2d6ef66c0062d882cee901afdb078cec274aaf632f8eae5ba5c44d8cf76"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/bundle.json","state_url":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/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":"2026-07-01T07:54:13Z","links":{"resolver":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY","bundle":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/bundle.json","state":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7R6YTMXW2CDGAAL3ELIETBZNDY","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":"3c35287c4854217b9ac3f98b164654abcb16fed55800a3d18d6327ad45d5e5d2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-14T13:23:33Z","title_canon_sha256":"156304b26d0e0c545d7cbedd977ba621a5053c3529b156f425163dc8ad56923f"},"schema_version":"1.0","source":{"id":"1308.3115","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.3115","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"arxiv_version","alias_value":"1308.3115v2","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.3115","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"pith_short_12","alias_value":"7R6YTMXW2CDG","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7R6YTMXW2CDGAAL3","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7R6YTMXW","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:150ab2d6ef66c0062d882cee901afdb078cec274aaf632f8eae5ba5c44d8cf76","target":"graph","created_at":"2026-05-18T03:15:55Z","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":"We study the following coupled Schr\\\"odinger system \\ds -\\Delta u+u=u^{2^*-1}+\\be u^{\\frac{2^*}{2}-1}v^{\\frac{2^*}{2}}+\\la_1u^{\\al-1}, &x\\in \\R^N, \\ds -\\Delta v+v=v^{2^*-1}+\\be u^{\\frac{2^*}{2}}v^{\\frac{2^*}{2}-1}+\\la_2v^{r-1}, &x\\in \\R^N, u,v > 0, &x\\in \\R^N, where $N\\geq 5, \\la_1,\\la_2>0,\\be\\neq 0, 2<\\al,r<2^*,2^*\\triangleq \\frac{2N}{N-2}.$ Note that the nonlinearity and the coupling terms are both critical. Using the Mountain Pass Theorem, Ekeland's variational principle and Nehari mainfold, we show that this critical system has a positive radial solution for positive $\\be$ and some negativ","authors_text":"Hong-yu Ye, Yan-fang Peng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-14T13:23:33Z","title":"Positive radial solutions for coupled Schr\\\"{o}dinger system with critical exponent in $\\R^N\\,(N\\geq5)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3115","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:ecbea8aa3987ba15ed509b13be6b4c160b7230bd62ac4daaaa11112b5b2e0d3f","target":"record","created_at":"2026-05-18T03:15:55Z","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":"3c35287c4854217b9ac3f98b164654abcb16fed55800a3d18d6327ad45d5e5d2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-14T13:23:33Z","title_canon_sha256":"156304b26d0e0c545d7cbedd977ba621a5053c3529b156f425163dc8ad56923f"},"schema_version":"1.0","source":{"id":"1308.3115","kind":"arxiv","version":2}},"canonical_sha256":"fc7d89b2f6d08660017b22d049872d1e3b1e388d318b9d43b8c332b91f0cd875","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc7d89b2f6d08660017b22d049872d1e3b1e388d318b9d43b8c332b91f0cd875","first_computed_at":"2026-05-18T03:15:55.347731Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:55.347731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H4DTOSIMIrmdS/0o7vTrj4kTEOZinGhVM9zWwLJnAAGLqYobGX4jvevBtKInCs19/PQ7dbdzMusyY0MAlF4dAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:55.348517Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.3115","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ecbea8aa3987ba15ed509b13be6b4c160b7230bd62ac4daaaa11112b5b2e0d3f","sha256:150ab2d6ef66c0062d882cee901afdb078cec274aaf632f8eae5ba5c44d8cf76"],"state_sha256":"03a8a77b9dac42e4cf2e0213b5308717a8a6a63351188437cb55452d9a9551f3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6TaWULdePgLC+djZRK4RIRS0/Kx6unEA3KSGjKe47SGfH4z8RAsfDGvTnmqahTrOrK2K2KViyXPnY8yAfW39Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T07:54:13.387429Z","bundle_sha256":"65bd618589fa321ea7e6e6460183c5fb051bfb7be9c55c344e44541c9d3c5283"}}