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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_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1504.01005","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-04T10:54:39Z","cross_cats_sorted":[],"title_canon_sha256":"369c38c82a5093d82ebbf6d069a0de59d937a36377d2cf62b037c9c978764f64","abstract_canon_sha256":"259f2870002fd1699845301e326f6ee963b886b51eaaf5f6e54a688177acd9ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:13.459697Z","signature_b64":"zO0R//T9D5KyyLYSga9VDF4r3O6nbTBuQaaxnnmF/D9wuvn2BqRdtBN9aNpjL7imudGKFBJiZIiP0ackpVGQAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d0cd9c4224a483a69afd1ab622744216df37785f15d3311b41decd17b2fcb225","last_reissued_at":"2026-05-18T01:37:13.459257Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:13.459257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Elliptic Systems involving critical Hardy-Sobolev exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-04T10:54:39Z","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_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01005","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.01005","created_at":"2026-05-18T01:37:13.459322+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.01005v2","created_at":"2026-05-18T01:37:13.459322+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.01005","created_at":"2026-05-18T01:37:13.459322+00:00"},{"alias_kind":"pith_short_12","alias_value":"2DGZYQREUSB2","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_16","alias_value":"2DGZYQREUSB2NGX5","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_8","alias_value":"2DGZYQRE","created_at":"2026-05-18T12:28:59.999130+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3","json":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3.json","graph_json":"https://pith.science/api/pith-number/2DGZYQREUSB2NGX5DK3CE5CCC3/graph.json","events_json":"https://pith.science/api/pith-number/2DGZYQREUSB2NGX5DK3CE5CCC3/events.json","paper":"https://pith.science/paper/2DGZYQRE"},"agent_actions":{"view_html":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3","download_json":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3.json","view_paper":"https://pith.science/paper/2DGZYQRE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.01005&json=true","fetch_graph":"https://pith.science/api/pith-number/2DGZYQREUSB2NGX5DK3CE5CCC3/graph.json","fetch_events":"https://pith.science/api/pith-number/2DGZYQREUSB2NGX5DK3CE5CCC3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3/action/storage_attestation","attest_author":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3/action/author_attestation","sign_citation":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3/action/citation_signature","submit_replication":"https://pith.science/pith/2DGZYQREUSB2NGX5DK3CE5CCC3/action/replication_record"}},"created_at":"2026-05-18T01:37:13.459322+00:00","updated_at":"2026-05-18T01:37:13.459322+00:00"}