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We show the extremal solutions associated with the above system are bounded provided\n  [\\frac{N}{2} < 1 + \\frac{2(\\theta+1)}{p\\theta -1} (\\sqrt{\\frac{p \\theta (p+1)}{\\theta +1}} + \\sqrt{\\frac{p \\theta (p+1)}{\\theta +1} - \\sqrt{\\frac{p \\theta (p+1)}{\\theta +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":"1206.4273","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-19T17:21:44Z","cross_cats_sorted":[],"title_canon_sha256":"608ace14edc14c40ac1c0e24695d464d1fc243a589131ffd37ee8d95d2b47029","abstract_canon_sha256":"2a339a8f9512e3e13b41a0f7e33d01234d8239a26cef7e60f2f5d8edcf2c64ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:07.536820Z","signature_b64":"wz0dUk5Hi0Z6cq4/OjJoPCvDURrtrDu1bq8aLyWS8TV7wM7VuFAGGc5TKTZvHvGhSOGzx1ki7fsPWEugKUPEBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74824b97393aece5b5cbc02a413c4879fdcb14286b2f94e549fda31870bbf9bc","last_reissued_at":"2026-05-18T03:53:07.536126Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:07.536126Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of stable solutions of a Lane-Emden type system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan","submitted_at":"2012-06-19T17:21:44Z","abstract_excerpt":"We examine the system given by\n\\hfill -\\Delta u = \\lambda (v+1)^p \\qquad \\Omega\n\\hfill -\\Delta v = \\gamma (u+1)^\\theta \\qquad \\Omega,\n\\hfill u = v =0 \\qquad \\quad \\partial \\Omega,\nwhere $ \\lambda,\\gamma$ are positive parameters and where $ 1 < p \\le \\theta$ and where $ \\Omega$ is a smooth bounded domain in $ R^N$. We show the extremal solutions associated with the above system are bounded provided\n  [\\frac{N}{2} < 1 + \\frac{2(\\theta+1)}{p\\theta -1} (\\sqrt{\\frac{p \\theta (p+1)}{\\theta +1}} + \\sqrt{\\frac{p \\theta (p+1)}{\\theta +1} - \\sqrt{\\frac{p \\theta (p+1)}{\\theta +1}}})]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4273","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":""},"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":"1206.4273","created_at":"2026-05-18T03:53:07.536226+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.4273v1","created_at":"2026-05-18T03:53:07.536226+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4273","created_at":"2026-05-18T03:53:07.536226+00:00"},{"alias_kind":"pith_short_12","alias_value":"OSBEXFZZHLWO","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"OSBEXFZZHLWOLNOL","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"OSBEXFZZ","created_at":"2026-05-18T12:27:16.716162+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/OSBEXFZZHLWOLNOLYAVECPCIPH","json":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH.json","graph_json":"https://pith.science/api/pith-number/OSBEXFZZHLWOLNOLYAVECPCIPH/graph.json","events_json":"https://pith.science/api/pith-number/OSBEXFZZHLWOLNOLYAVECPCIPH/events.json","paper":"https://pith.science/paper/OSBEXFZZ"},"agent_actions":{"view_html":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH","download_json":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH.json","view_paper":"https://pith.science/paper/OSBEXFZZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.4273&json=true","fetch_graph":"https://pith.science/api/pith-number/OSBEXFZZHLWOLNOLYAVECPCIPH/graph.json","fetch_events":"https://pith.science/api/pith-number/OSBEXFZZHLWOLNOLYAVECPCIPH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH/action/storage_attestation","attest_author":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH/action/author_attestation","sign_citation":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH/action/citation_signature","submit_replication":"https://pith.science/pith/OSBEXFZZHLWOLNOLYAVECPCIPH/action/replication_record"}},"created_at":"2026-05-18T03:53:07.536226+00:00","updated_at":"2026-05-18T03:53:07.536226+00:00"}