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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"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1308.3115","created_at":"2026-05-18T03:15:55.347842+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.3115v2","created_at":"2026-05-18T03:15:55.347842+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.3115","created_at":"2026-05-18T03:15:55.347842+00:00"},{"alias_kind":"pith_short_12","alias_value":"7R6YTMXW2CDG","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"7R6YTMXW2CDGAAL3","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"7R6YTMXW","created_at":"2026-05-18T12:27:36.564083+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/7R6YTMXW2CDGAAL3ELIETBZNDY","json":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY.json","graph_json":"https://pith.science/api/pith-number/7R6YTMXW2CDGAAL3ELIETBZNDY/graph.json","events_json":"https://pith.science/api/pith-number/7R6YTMXW2CDGAAL3ELIETBZNDY/events.json","paper":"https://pith.science/paper/7R6YTMXW"},"agent_actions":{"view_html":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY","download_json":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY.json","view_paper":"https://pith.science/paper/7R6YTMXW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.3115&json=true","fetch_graph":"https://pith.science/api/pith-number/7R6YTMXW2CDGAAL3ELIETBZNDY/graph.json","fetch_events":"https://pith.science/api/pith-number/7R6YTMXW2CDGAAL3ELIETBZNDY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/action/storage_attestation","attest_author":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/action/author_attestation","sign_citation":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/action/citation_signature","submit_replication":"https://pith.science/pith/7R6YTMXW2CDGAAL3ELIETBZNDY/action/replication_record"}},"created_at":"2026-05-18T03:15:55.347842+00:00","updated_at":"2026-05-18T03:15:55.347842+00:00"}