{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:KQV3DJ7M6YCE67YYHOVGIREFXY","short_pith_number":"pith:KQV3DJ7M","schema_version":"1.0","canonical_sha256":"542bb1a7ecf6044f7f183baa644485be3171048b5701d74c54766c4d4af2a87a","source":{"kind":"arxiv","id":"2104.04158","version":1},"attestation_state":"computed","paper":{"title":"Normalized solutions to Schr\\\"{o}dinger systems with linear and nonlinear couplings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Zhaoyang Yun, Zhitao Zhang","submitted_at":"2021-04-09T02:21:47Z","abstract_excerpt":"In this paper, we study important Schr\\\"{o}dinger systems with linear and nonlinear couplings \\begin{equation}\\label{eq:diricichlet} \\begin{cases} -\\Delta u_1-\\lambda_1 u_1=\\mu_1 |u_1|^{p_1-2}u_1+r_1\\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\\kappa (x)u_2~\\hbox{in}~\\mathbb{R}^N,\\\\ -\\Delta u_2-\\lambda_2 u_2=\\mu_2 |u_2|^{p_2-2}u_2+r_2\\beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+\\kappa (x)u_1~ \\hbox{in}~\\mathbb{R}^N,\\\\ u_1\\in H^1(\\mathbb{R}^N), u_2\\in H^1(\\mathbb{R}^N),\\nonumber \\end{cases} \\end{equation} with the condition $$\\int_{\\mathbb{R}^N} u_1^2=a_1^2, \\int_{\\mathbb{R}^N} u_2^2=a_2^2,$$ where $N\\geq 2$, $\\mu_1"},"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":"2104.04158","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-04-09T02:21:47Z","cross_cats_sorted":[],"title_canon_sha256":"191509b2202635ae6cefedeee378040b750c315c4429047ab2637f861e1ec032","abstract_canon_sha256":"a6714e14962c79e3adc10d2ff57b2c3a001341744a94e41f744f250c229e82cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:30:36.238059Z","signature_b64":"argmXJoSWhPdXjcmdaeakS6JUp39CrxQHlHqmwQs51mit1dGjntZJrUFlXoX4neRdcb26QWYzoQ1/SLJGFVxDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"542bb1a7ecf6044f7f183baa644485be3171048b5701d74c54766c4d4af2a87a","last_reissued_at":"2026-07-05T02:30:36.237652Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:30:36.237652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normalized solutions to Schr\\\"{o}dinger systems with linear and nonlinear couplings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Zhaoyang Yun, Zhitao Zhang","submitted_at":"2021-04-09T02:21:47Z","abstract_excerpt":"In this paper, we study important Schr\\\"{o}dinger systems with linear and nonlinear couplings \\begin{equation}\\label{eq:diricichlet} \\begin{cases} -\\Delta u_1-\\lambda_1 u_1=\\mu_1 |u_1|^{p_1-2}u_1+r_1\\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\\kappa (x)u_2~\\hbox{in}~\\mathbb{R}^N,\\\\ -\\Delta u_2-\\lambda_2 u_2=\\mu_2 |u_2|^{p_2-2}u_2+r_2\\beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+\\kappa (x)u_1~ \\hbox{in}~\\mathbb{R}^N,\\\\ u_1\\in H^1(\\mathbb{R}^N), u_2\\in H^1(\\mathbb{R}^N),\\nonumber \\end{cases} \\end{equation} with the condition $$\\int_{\\mathbb{R}^N} u_1^2=a_1^2, \\int_{\\mathbb{R}^N} u_2^2=a_2^2,$$ where $N\\geq 2$, $\\mu_1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.04158","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2104.04158/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2104.04158","created_at":"2026-07-05T02:30:36.237717+00:00"},{"alias_kind":"arxiv_version","alias_value":"2104.04158v1","created_at":"2026-07-05T02:30:36.237717+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.04158","created_at":"2026-07-05T02:30:36.237717+00:00"},{"alias_kind":"pith_short_12","alias_value":"KQV3DJ7M6YCE","created_at":"2026-07-05T02:30:36.237717+00:00"},{"alias_kind":"pith_short_16","alias_value":"KQV3DJ7M6YCE67YY","created_at":"2026-07-05T02:30:36.237717+00:00"},{"alias_kind":"pith_short_8","alias_value":"KQV3DJ7M","created_at":"2026-07-05T02:30:36.237717+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/KQV3DJ7M6YCE67YYHOVGIREFXY","json":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY.json","graph_json":"https://pith.science/api/pith-number/KQV3DJ7M6YCE67YYHOVGIREFXY/graph.json","events_json":"https://pith.science/api/pith-number/KQV3DJ7M6YCE67YYHOVGIREFXY/events.json","paper":"https://pith.science/paper/KQV3DJ7M"},"agent_actions":{"view_html":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY","download_json":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY.json","view_paper":"https://pith.science/paper/KQV3DJ7M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2104.04158&json=true","fetch_graph":"https://pith.science/api/pith-number/KQV3DJ7M6YCE67YYHOVGIREFXY/graph.json","fetch_events":"https://pith.science/api/pith-number/KQV3DJ7M6YCE67YYHOVGIREFXY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY/action/storage_attestation","attest_author":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY/action/author_attestation","sign_citation":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY/action/citation_signature","submit_replication":"https://pith.science/pith/KQV3DJ7M6YCE67YYHOVGIREFXY/action/replication_record"}},"created_at":"2026-07-05T02:30:36.237717+00:00","updated_at":"2026-07-05T02:30:36.237717+00:00"}