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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","authors_text":"Mostafa Fazly, Nassif Ghoussoub","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-07-28T00:30:16Z","title":"On the H\\'enon-Lane-Emden conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5611","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5e88fb354ac4c38dbc9050b2584a95f4ceed336e9fa8542159abde1e93d70175","target":"record","created_at":"2026-05-18T03:44:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"df0eb419204d7e7cfec11c8969ce54d007ef73fa26116dc8058edae398779c88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-07-28T00:30:16Z","title_canon_sha256":"633467e28b3c7c3a31a1c642de73c98ce76cdbe4fb66c9c5ce143c1501629e00"},"schema_version":"1.0","source":{"id":"1107.5611","kind":"arxiv","version":2}},"canonical_sha256":"280feeee21ee6e31608336cf008b9d800841f45917d7a4ef4c768fcadc4a82b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"280feeee21ee6e31608336cf008b9d800841f45917d7a4ef4c768fcadc4a82b0","first_computed_at":"2026-05-18T03:44:33.935905Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:33.935905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OZSVbfkEpeyS41wU3uEmi6ricOhKL2KRvUE1i4uGi3gv8X0qA5cRueAY+GAqQ0lh9+5aySfd+Xjnn2KGZeGYBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:33.936638Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.5611","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e88fb354ac4c38dbc9050b2584a95f4ceed336e9fa8542159abde1e93d70175","sha256:4931c8341fdc4dcbf258a6947d00d44bbe16da745702e54d08be58cd6846d92b"],"state_sha256":"12b7c4f58d41d4456e6dfc647fbdba500421f91bce63257b590f031f45bfc1fd"}