{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:XJOJNKYH3VZJIRLIKR2DUUF7V7","short_pith_number":"pith:XJOJNKYH","schema_version":"1.0","canonical_sha256":"ba5c96ab07dd7294456854743a50bfafd26556e8df86b045dce253b4c692f9e3","source":{"kind":"arxiv","id":"1708.09701","version":2},"attestation_state":"computed","paper":{"title":"Existence and phase separation of entire solutions to a pure critical competitive elliptic system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, M\\'onica Clapp","submitted_at":"2017-08-31T13:31:34Z","abstract_excerpt":"We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \\[ \\left\\{ \\begin{tabular} [c]{l}% $-\\Delta u=\\mu_{1}|u|^{{2}^{\\ast}-2}u+\\lambda\\alpha|u|^{\\alpha-2}|v|^{\\beta }u,$\\\\ $-\\Delta v=\\mu_{2}|v|^{{2}^{\\ast}-2}v+\\lambda\\beta|u|^{\\alpha}|v|^{\\beta{-2}% }v,$\\\\ $u,v\\in D^{1,2}(\\mathbb{R}^{N}),$% \\end{tabular} \\ \\right. \\] where $N\\geq4,$ $2^{\\ast}:=\\frac{2N}{N-2}$ is the critical Sobolev exponent, $\\alpha,\\beta\\in(1,2],$ $\\alpha+\\beta=2^{\\ast},$ $\\mu_{1},\\mu_{2}>0,$ and $\\lambda<0.$ We show that these solutions exhibit phase separation as "},"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":"1708.09701","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-08-31T13:31:34Z","cross_cats_sorted":[],"title_canon_sha256":"242d9ce992ea9be42f0811323d33ce2acba3a6b3130f778cb6eaa460bb775ee9","abstract_canon_sha256":"818a8340023c96bbe5d5b457344716429928d0dc7020131e79ac8cefd31b4b0f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:41.614949Z","signature_b64":"PJU+PWI05dN4iYRJuvIOeFzaS2g2p6vuWJfLlG7PigNX7YrTGyYxWqNO1mMAOU/dTzAJQXQR3HpnvS9rMi/tAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba5c96ab07dd7294456854743a50bfafd26556e8df86b045dce253b4c692f9e3","last_reissued_at":"2026-05-18T00:30:41.614569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:41.614569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence and phase separation of entire solutions to a pure critical competitive elliptic system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, M\\'onica Clapp","submitted_at":"2017-08-31T13:31:34Z","abstract_excerpt":"We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \\[ \\left\\{ \\begin{tabular} [c]{l}% $-\\Delta u=\\mu_{1}|u|^{{2}^{\\ast}-2}u+\\lambda\\alpha|u|^{\\alpha-2}|v|^{\\beta }u,$\\\\ $-\\Delta v=\\mu_{2}|v|^{{2}^{\\ast}-2}v+\\lambda\\beta|u|^{\\alpha}|v|^{\\beta{-2}% }v,$\\\\ $u,v\\in D^{1,2}(\\mathbb{R}^{N}),$% \\end{tabular} \\ \\right. \\] where $N\\geq4,$ $2^{\\ast}:=\\frac{2N}{N-2}$ is the critical Sobolev exponent, $\\alpha,\\beta\\in(1,2],$ $\\alpha+\\beta=2^{\\ast},$ $\\mu_{1},\\mu_{2}>0,$ and $\\lambda<0.$ We show that these solutions exhibit phase separation as "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.09701","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":"1708.09701","created_at":"2026-05-18T00:30:41.614630+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.09701v2","created_at":"2026-05-18T00:30:41.614630+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.09701","created_at":"2026-05-18T00:30:41.614630+00:00"},{"alias_kind":"pith_short_12","alias_value":"XJOJNKYH3VZJ","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"XJOJNKYH3VZJIRLI","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"XJOJNKYH","created_at":"2026-05-18T12:31:53.515858+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/XJOJNKYH3VZJIRLIKR2DUUF7V7","json":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7.json","graph_json":"https://pith.science/api/pith-number/XJOJNKYH3VZJIRLIKR2DUUF7V7/graph.json","events_json":"https://pith.science/api/pith-number/XJOJNKYH3VZJIRLIKR2DUUF7V7/events.json","paper":"https://pith.science/paper/XJOJNKYH"},"agent_actions":{"view_html":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7","download_json":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7.json","view_paper":"https://pith.science/paper/XJOJNKYH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.09701&json=true","fetch_graph":"https://pith.science/api/pith-number/XJOJNKYH3VZJIRLIKR2DUUF7V7/graph.json","fetch_events":"https://pith.science/api/pith-number/XJOJNKYH3VZJIRLIKR2DUUF7V7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7/action/storage_attestation","attest_author":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7/action/author_attestation","sign_citation":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7/action/citation_signature","submit_replication":"https://pith.science/pith/XJOJNKYH3VZJIRLIKR2DUUF7V7/action/replication_record"}},"created_at":"2026-05-18T00:30:41.614630+00:00","updated_at":"2026-05-18T00:30:41.614630+00:00"}