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We study positive solutions of equation (E) $-L_\\mu u+ g(|\\nabla u|) = 0$ in $\\Omega$ where $L_\\mu=\\Delta + \\frac{\\mu}{\\delta^2} $, $\\mu \\in (0,\\frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${\\mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $\\nu$. When $g(t)=t^q$ with $q \\in (1,2)$, we show that equation (E) admits a critical exponent $q_\\mu$ (dependi"},"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":"1903.11090","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-26T18:02:18Z","cross_cats_sorted":[],"title_canon_sha256":"146b396c44bc2e532872019b686f5f25839c829fc7d5ed3efc75963ffd0b3636","abstract_canon_sha256":"30e0fc2c455d31716755afdfb3e1beddd680567aed66e012cfef15846f35a02c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:04.201137Z","signature_b64":"80xqimpTeJdSEIygCHZY2tTQHY5zQ6p/GqVEP20Z2sZQlqQDDYDCCAwxPsmel7R+Ep9jxSuTAJxkZXepSmeVBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98dc9efd7ce838ae0d309413580f197385cd302ad25434016ca688750f13ddc7","last_reissued_at":"2026-05-17T23:50:04.200615Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:04.200615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semilinear elliptic equations with Hardy potential and gradient nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Konstantinos Gkikas, Phuoc-Tai Nguyen","submitted_at":"2019-03-26T18:02:18Z","abstract_excerpt":"Let $\\Omega \\subset {\\mathbb R}^N$ ($N \\geq 3$) be a $C^2$ bounded domain and $\\delta$ be the distance to $\\partial \\Omega$. 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When $g(t)=t^q$ with $q \\in (1,2)$, we show that equation (E) admits a critical exponent $q_\\mu$ (dependi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.11090","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":"1903.11090","created_at":"2026-05-17T23:50:04.200691+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.11090v1","created_at":"2026-05-17T23:50:04.200691+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.11090","created_at":"2026-05-17T23:50:04.200691+00:00"},{"alias_kind":"pith_short_12","alias_value":"TDOJ57L45A4K","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_16","alias_value":"TDOJ57L45A4K4DJQ","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_8","alias_value":"TDOJ57L4","created_at":"2026-05-18T12:33:27.125529+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/TDOJ57L45A4K4DJQSQJVQDYZOO","json":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO.json","graph_json":"https://pith.science/api/pith-number/TDOJ57L45A4K4DJQSQJVQDYZOO/graph.json","events_json":"https://pith.science/api/pith-number/TDOJ57L45A4K4DJQSQJVQDYZOO/events.json","paper":"https://pith.science/paper/TDOJ57L4"},"agent_actions":{"view_html":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO","download_json":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO.json","view_paper":"https://pith.science/paper/TDOJ57L4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.11090&json=true","fetch_graph":"https://pith.science/api/pith-number/TDOJ57L45A4K4DJQSQJVQDYZOO/graph.json","fetch_events":"https://pith.science/api/pith-number/TDOJ57L45A4K4DJQSQJVQDYZOO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO/action/storage_attestation","attest_author":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO/action/author_attestation","sign_citation":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO/action/citation_signature","submit_replication":"https://pith.science/pith/TDOJ57L45A4K4DJQSQJVQDYZOO/action/replication_record"}},"created_at":"2026-05-17T23:50:04.200691+00:00","updated_at":"2026-05-17T23:50:04.200691+00:00"}