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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 $p+q\\<1+\\frac{4}{n-1}$.","authors_text":"Laurent Veron (LMPT), Marie-Fran\\c{c}oise Bidaut-Veron (LMPT), Marta Garcia-Huidobro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-30T16:14:35Z","title":"Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11489","kind":"arxiv","version":3},"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:cd0c3d36b436375c6a0178c96b8257563aa4dfc16f3d1ff0d4081bd2b540c984","target":"record","created_at":"2026-05-17T23:44:59Z","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":"621cdbb22cf52eca458879eafdfca61e4ca010d5bd82eb7295cd11ebc1e4328f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-30T16:14:35Z","title_canon_sha256":"d59b840f8f5e16df3fc91b5114f7a78e7735d78f312e2a189b51f4f22882e659"},"schema_version":"1.0","source":{"id":"1711.11489","kind":"arxiv","version":3}},"canonical_sha256":"62ce91f478d13e82ac6458b3852fe163244d5365be8b433c5a5c5cf81a0161c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62ce91f478d13e82ac6458b3852fe163244d5365be8b433c5a5c5cf81a0161c2","first_computed_at":"2026-05-17T23:44:59.944563Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:59.944563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3AkkHG+Z84qZCTPWJVfO6W2LXGBt90keslWz8CKwLCu3l3eIOCqnf2ik4qdEFMeYN7DtMoQXfnr51kRF+7pxBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:59.945253Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.11489","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cd0c3d36b436375c6a0178c96b8257563aa4dfc16f3d1ff0d4081bd2b540c984","sha256:c66e5ddad3598fad869f298a35746e706c235efd0ceffbd71efd5f4f44f56782"],"state_sha256":"e2158ec8860f7ee96a489009345aae38603e2b0155ea6c2c1fdd8c1d3fe71fa5"}