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A positive solution of (E) is moderate if it is dominated by an $L_\\mu$-harmonic function. If $\\mu<C_H(\\Omega)$ (the Hardy constant for $\\Omega$) every positive $L_\\mu$- harmonic functions can be represented in terms of a finite measure on $\\partial\\Omega$ via the Martin representation theorem. However the classical measure boundary trace of any such"},"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":"1407.3572","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-14T09:08:34Z","cross_cats_sorted":[],"title_canon_sha256":"30004b8ad000d9ffeb69230cc657d9effe9ba7a04e9c28143422a8784ee028b2","abstract_canon_sha256":"94217e24745c1a0dfc87dc17c9e89636405ecbf76b6e5669202c46d2ae7dcda0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:42.443284Z","signature_b64":"jAGhM6o5dsJoRzR0oph+ayc7Pk3c84N6ZmppYpnMV/+5vZuWTE8x8kH+ZnDf0zOhRIrj4FomfYQtAa8REyroDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"172326dc790589244f502cd061f33323702536358320b4ab5730a50e404cce7d","last_reissued_at":"2026-05-18T02:47:42.442764Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:42.442764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Moderate solutions of semilinear elliptic equations with Hardy potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Moshe Marcus, Phuoc-Tai Nguyen","submitted_at":"2014-07-14T09:08:34Z","abstract_excerpt":"Let $\\Omega$ be a bounded smooth domain in $\\mathbb{R}^N$. We study positive solutions of equation (E) $-L_\\mu u+ u^q = 0$ in $\\Omega$ where $L_\\mu=\\Delta + \\frac{\\mu}{\\delta^2}$,\n  $0<\\mu$, $q>1$ and $\\delta(x)=\\mathrm{dist}\\,(x,\\partial\\Omega)$. A positive solution of (E) is moderate if it is dominated by an $L_\\mu$-harmonic function. If $\\mu<C_H(\\Omega)$ (the Hardy constant for $\\Omega$) every positive $L_\\mu$- harmonic functions can be represented in terms of a finite measure on $\\partial\\Omega$ via the Martin representation theorem. However the classical measure boundary trace of any such"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3572","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":"1407.3572","created_at":"2026-05-18T02:47:42.442845+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.3572v1","created_at":"2026-05-18T02:47:42.442845+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.3572","created_at":"2026-05-18T02:47:42.442845+00:00"},{"alias_kind":"pith_short_12","alias_value":"C4RSNXDZAWES","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"C4RSNXDZAWESIT2Q","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"C4RSNXDZ","created_at":"2026-05-18T12:28:22.404517+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/C4RSNXDZAWESIT2QFTIGD4ZTEN","json":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN.json","graph_json":"https://pith.science/api/pith-number/C4RSNXDZAWESIT2QFTIGD4ZTEN/graph.json","events_json":"https://pith.science/api/pith-number/C4RSNXDZAWESIT2QFTIGD4ZTEN/events.json","paper":"https://pith.science/paper/C4RSNXDZ"},"agent_actions":{"view_html":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN","download_json":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN.json","view_paper":"https://pith.science/paper/C4RSNXDZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.3572&json=true","fetch_graph":"https://pith.science/api/pith-number/C4RSNXDZAWESIT2QFTIGD4ZTEN/graph.json","fetch_events":"https://pith.science/api/pith-number/C4RSNXDZAWESIT2QFTIGD4ZTEN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN/action/storage_attestation","attest_author":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN/action/author_attestation","sign_citation":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN/action/citation_signature","submit_replication":"https://pith.science/pith/C4RSNXDZAWESIT2QFTIGD4ZTEN/action/replication_record"}},"created_at":"2026-05-18T02:47:42.442845+00:00","updated_at":"2026-05-18T02:47:42.442845+00:00"}