{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2DGZYQREUSB2NGX5DK3CE5CCC3","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":"259f2870002fd1699845301e326f6ee963b886b51eaaf5f6e54a688177acd9ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-04T10:54:39Z","title_canon_sha256":"369c38c82a5093d82ebbf6d069a0de59d937a36377d2cf62b037c9c978764f64"},"schema_version":"1.0","source":{"id":"1504.01005","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.01005","created_at":"2026-05-18T01:37:13Z"},{"alias_kind":"arxiv_version","alias_value":"1504.01005v2","created_at":"2026-05-18T01:37:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.01005","created_at":"2026-05-18T01:37:13Z"},{"alias_kind":"pith_short_12","alias_value":"2DGZYQREUSB2","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"2DGZYQREUSB2NGX5","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"2DGZYQRE","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:92e2554795bcf51ff0dcdd693e3d0aa119079fdda74e96363047e1dfd7049c98","target":"graph","created_at":"2026-05-18T01:37:13Z","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":"Let $\\Omega\\subset \\R^N$ ($N\\geq 3$) be an open domain which is not necessarily bounded. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\\begin{cases} -\\Delta u-\\lambda \\frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\\kappa\\alpha \\frac{1}{|x|^{s_2}}|u|^{\\alpha-2}u|v|^\\beta\\quad &\\hbox{in}\\;\\Omega,\\\\ -\\Delta v-\\mu \\frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=\\kappa\\beta \\frac{1}{|x|^{s_2}}|u|^{\\alpha}|v|^{\\beta-2}v\\quad &\\hbox{in}\\;\\Omega,\\\\ (u,v)\\in \\mathscr{D}:=D_{0}^{1,2}(\\Omega)\\times D_{0}^{1,2}(\\Omega), \\end{cases}$$ where $s_1,s_","authors_text":"Wenming Zou, Xuexiu Zhong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-04T10:54:39Z","title":"On Elliptic Systems involving critical Hardy-Sobolev exponents"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01005","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":"sha256:9f7f7c7ab0725e099b583986c1930e1240e34ea1ec1a36111935dde7b7c29c66","target":"record","created_at":"2026-05-18T01:37:13Z","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":"259f2870002fd1699845301e326f6ee963b886b51eaaf5f6e54a688177acd9ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-04T10:54:39Z","title_canon_sha256":"369c38c82a5093d82ebbf6d069a0de59d937a36377d2cf62b037c9c978764f64"},"schema_version":"1.0","source":{"id":"1504.01005","kind":"arxiv","version":2}},"canonical_sha256":"d0cd9c4224a483a69afd1ab622744216df37785f15d3311b41decd17b2fcb225","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d0cd9c4224a483a69afd1ab622744216df37785f15d3311b41decd17b2fcb225","first_computed_at":"2026-05-18T01:37:13.459257Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:13.459257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zO0R//T9D5KyyLYSga9VDF4r3O6nbTBuQaaxnnmF/D9wuvn2BqRdtBN9aNpjL7imudGKFBJiZIiP0ackpVGQAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:13.459697Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.01005","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9f7f7c7ab0725e099b583986c1930e1240e34ea1ec1a36111935dde7b7c29c66","sha256:92e2554795bcf51ff0dcdd693e3d0aa119079fdda74e96363047e1dfd7049c98"],"state_sha256":"e3504dd4a8fdcc9c5428cf6b492da87ea8f23f4c03acc059aeb3a9259b3c53ce"}