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The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \\frac{N+a}{p+1}+\\frac{N+b}{q+1}>{N-2}$.\n  We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case.\n  Assumi"},"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":"1107.5611","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-07-28T00:30:16Z","cross_cats_sorted":[],"title_canon_sha256":"633467e28b3c7c3a31a1c642de73c98ce76cdbe4fb66c9c5ce143c1501629e00","abstract_canon_sha256":"df0eb419204d7e7cfec11c8969ce54d007ef73fa26116dc8058edae398779c88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:33.936638Z","signature_b64":"OZSVbfkEpeyS41wU3uEmi6ricOhKL2KRvUE1i4uGi3gv8X0qA5cRueAY+GAqQ0lh9+5aySfd+Xjnn2KGZeGYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"280feeee21ee6e31608336cf008b9d800841f45917d7a4ef4c768fcadc4a82b0","last_reissued_at":"2026-05-18T03:44:33.935905Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:33.935905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the H\\'enon-Lane-Emden conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly, Nassif Ghoussoub","submitted_at":"2011-07-28T00:30:16Z","abstract_excerpt":"We consider Liouville-type theorems for the following H\\'{e}non-Lane-Emden system\n  \\hfill -\\Delta u&=& |x|^{a}v^p \\text{in} \\mathbb{R}^N,\n  \\hfill -\\Delta v&=& |x|^{b}u^q \\text{in} \\mathbb{R}^N,\nwhen $pq>1$, $p,q,a,b\\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \\frac{N+a}{p+1}+\\frac{N+b}{q+1}>{N-2}$.\n  We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case.\n  Assumi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5611","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":"1107.5611","created_at":"2026-05-18T03:44:33.936018+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.5611v2","created_at":"2026-05-18T03:44:33.936018+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.5611","created_at":"2026-05-18T03:44:33.936018+00:00"},{"alias_kind":"pith_short_12","alias_value":"FAH653RB5ZXD","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FAH653RB5ZXDCYED","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FAH653RB","created_at":"2026-05-18T12:26:28.662955+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/FAH653RB5ZXDCYEDG3HQBC45QA","json":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA.json","graph_json":"https://pith.science/api/pith-number/FAH653RB5ZXDCYEDG3HQBC45QA/graph.json","events_json":"https://pith.science/api/pith-number/FAH653RB5ZXDCYEDG3HQBC45QA/events.json","paper":"https://pith.science/paper/FAH653RB"},"agent_actions":{"view_html":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA","download_json":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA.json","view_paper":"https://pith.science/paper/FAH653RB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.5611&json=true","fetch_graph":"https://pith.science/api/pith-number/FAH653RB5ZXDCYEDG3HQBC45QA/graph.json","fetch_events":"https://pith.science/api/pith-number/FAH653RB5ZXDCYEDG3HQBC45QA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA/action/storage_attestation","attest_author":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA/action/author_attestation","sign_citation":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA/action/citation_signature","submit_replication":"https://pith.science/pith/FAH653RB5ZXDCYEDG3HQBC45QA/action/replication_record"}},"created_at":"2026-05-18T03:44:33.936018+00:00","updated_at":"2026-05-18T03:44:33.936018+00:00"}