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We first prove a local Harnack inequality and nonexistence of positive solutions in ${\\mathbb R}^N$ when $p(N-2)+q(N-1) \\<N$ or in an exterior domain if $p(N-2)+q(N-1)\\<N$ and $0\\leq q\\<1$. Using a direct Bernstein method we obtain a first range of values of $p$ and $q$ in which $u(x)\\leq c({\\mathrm dist\\,}(x,\\partial\\Omega)^{\\frac{q-2}{p+q-1}}$ This holds in particular if $p+q\\<1+\\frac{4}{n-1}$."},"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":"1711.11489","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-30T16:14:35Z","cross_cats_sorted":[],"title_canon_sha256":"d59b840f8f5e16df3fc91b5114f7a78e7735d78f312e2a189b51f4f22882e659","abstract_canon_sha256":"621cdbb22cf52eca458879eafdfca61e4ca010d5bd82eb7295cd11ebc1e4328f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:59.945253Z","signature_b64":"3AkkHG+Z84qZCTPWJVfO6W2LXGBt90keslWz8CKwLCu3l3eIOCqnf2ik4qdEFMeYN7DtMoQXfnr51kRF+7pxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62ce91f478d13e82ac6458b3852fe163244d5365be8b433c5a5c5cf81a0161c2","last_reissued_at":"2026-05-17T23:44:59.944563Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:59.944563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Laurent Veron (LMPT), Marie-Fran\\c{c}oise Bidaut-Veron (LMPT), Marta Garcia-Huidobro","submitted_at":"2017-11-30T16:14:35Z","abstract_excerpt":"We study local and global properties of positive solutions of $-{\\Delta}u=u^p]{\\left |{\\nabla u}\\right |}^q$ in a domain ${\\Omega}$ of ${\\mathbb R}^N$, in the range $1\\<p+q$, $p\\geq 0$, $0\\leq q\\< 2$. We first prove a local Harnack inequality and nonexistence of positive solutions in ${\\mathbb R}^N$ when $p(N-2)+q(N-1) \\<N$ or in an exterior domain if $p(N-2)+q(N-1)\\<N$ and $0\\leq q\\<1$. Using a direct Bernstein method we obtain a first range of values of $p$ and $q$ in which $u(x)\\leq c({\\mathrm dist\\,}(x,\\partial\\Omega)^{\\frac{q-2}{p+q-1}}$ This holds in particular if $p+q\\<1+\\frac{4}{n-1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11489","kind":"arxiv","version":3},"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":"1711.11489","created_at":"2026-05-17T23:44:59.944677+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.11489v3","created_at":"2026-05-17T23:44:59.944677+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.11489","created_at":"2026-05-17T23:44:59.944677+00:00"},{"alias_kind":"pith_short_12","alias_value":"MLHJD5DY2E7I","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_16","alias_value":"MLHJD5DY2E7IFLDE","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_8","alias_value":"MLHJD5DY","created_at":"2026-05-18T12:31:31.346846+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/MLHJD5DY2E7IFLDELCZYKL7BMM","json":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM.json","graph_json":"https://pith.science/api/pith-number/MLHJD5DY2E7IFLDELCZYKL7BMM/graph.json","events_json":"https://pith.science/api/pith-number/MLHJD5DY2E7IFLDELCZYKL7BMM/events.json","paper":"https://pith.science/paper/MLHJD5DY"},"agent_actions":{"view_html":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM","download_json":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM.json","view_paper":"https://pith.science/paper/MLHJD5DY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.11489&json=true","fetch_graph":"https://pith.science/api/pith-number/MLHJD5DY2E7IFLDELCZYKL7BMM/graph.json","fetch_events":"https://pith.science/api/pith-number/MLHJD5DY2E7IFLDELCZYKL7BMM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM/action/storage_attestation","attest_author":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM/action/author_attestation","sign_citation":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM/action/citation_signature","submit_replication":"https://pith.science/pith/MLHJD5DY2E7IFLDELCZYKL7BMM/action/replication_record"}},"created_at":"2026-05-17T23:44:59.944677+00:00","updated_at":"2026-05-17T23:44:59.944677+00:00"}