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This problem is related to coupled nonlinear Schr\\\"{o}dinger equations with critical exponent for Bose-Einstein condensate. For different ranges of $N$, $\\al$, $\\bb$ and $"},"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":"1211.1514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-07T11:11:13Z","cross_cats_sorted":[],"title_canon_sha256":"35c1ca45c6ccd41459188f46bd8e551ae8362b5842e35e57f431a75c4193999c","abstract_canon_sha256":"d28d6b37b444e0e2119d04d2a6478e4147cab7398cf5a757b7c92bf6acbb63a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:18.664126Z","signature_b64":"SlhjlaJzvDl06d4UTZelW7fw+NMq1oH2QVeKMn+kJD0B53mCG7jDbrxTBxtZ6WpzFP3SsRuak6ZRMIrjeP+JBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8d65118b66d4a1a915e06210adeb365600bae3f405b13bc317486c1e5d2c10d","last_reissued_at":"2026-05-18T02:55:18.663689Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:18.663689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence and symmetry of positive ground states for a doubly critical Schrodinger system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Zhijie Chen","submitted_at":"2012-11-07T11:11:13Z","abstract_excerpt":"We study the following doubly critical Schr\\\"{o}dinger system $$-\\Delta u -\\frac{\\la_1}{|x|^2}u=u^{2^\\ast-1}+ \\nu \\al u^{\\al-1}v^\\bb, \\quad x\\in \\RN, -\\Delta v -\\frac{\\la_2}{|x|^2}v=v^{2^\\ast-1} + \\nu \\bb u^{\\al}v^{\\bb-1}, \\quad x\\in \\RN, u, v\\in D^{1, 2}(\\RN),\\quad u, v>0 in $\\RN\\setminus{0}$},$$ where $N\\ge 3$, $\\la_1, \\la_2\\in (0, \\frac{(N-2)^2}{4})$, $2^\\ast=\\frac{2N}{N-2}$ and $\\al>1, \\bb>1$ satisfying $\\al+\\bb=2^\\ast$. This problem is related to coupled nonlinear Schr\\\"{o}dinger equations with critical exponent for Bose-Einstein condensate. For different ranges of $N$, $\\al$, $\\bb$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1514","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":"1211.1514","created_at":"2026-05-18T02:55:18.663750+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.1514v2","created_at":"2026-05-18T02:55:18.663750+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1514","created_at":"2026-05-18T02:55:18.663750+00:00"},{"alias_kind":"pith_short_12","alias_value":"XDLFCGFWNVFB","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"XDLFCGFWNVFBVEK6","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"XDLFCGFW","created_at":"2026-05-18T12:27:27.928770+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/XDLFCGFWNVFBVEK6AYQQVXVTMV","json":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV.json","graph_json":"https://pith.science/api/pith-number/XDLFCGFWNVFBVEK6AYQQVXVTMV/graph.json","events_json":"https://pith.science/api/pith-number/XDLFCGFWNVFBVEK6AYQQVXVTMV/events.json","paper":"https://pith.science/paper/XDLFCGFW"},"agent_actions":{"view_html":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV","download_json":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV.json","view_paper":"https://pith.science/paper/XDLFCGFW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.1514&json=true","fetch_graph":"https://pith.science/api/pith-number/XDLFCGFWNVFBVEK6AYQQVXVTMV/graph.json","fetch_events":"https://pith.science/api/pith-number/XDLFCGFWNVFBVEK6AYQQVXVTMV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV/action/storage_attestation","attest_author":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV/action/author_attestation","sign_citation":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV/action/citation_signature","submit_replication":"https://pith.science/pith/XDLFCGFWNVFBVEK6AYQQVXVTMV/action/replication_record"}},"created_at":"2026-05-18T02:55:18.663750+00:00","updated_at":"2026-05-18T02:55:18.663750+00:00"}