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Let $\\ga_{\\pm}=1\\pm\\sqrt{1-4\\gk}$ be the two Hardy exponents, $\\gl_\\gk$ the first eigenvalue of $\\CL_\\gk$ with corresponding positive eigenfunction $\\phi_\\gk$. If $g$ is a continuous nondecreasing function satisfying $\\int_1^\\infty(g(s)+|g(-s)|)s^{-2\\frac{2N-2+\\ga_+}{2N-4+\\ga_+}}ds<\\infty$, then for any Radon measures $\\gn\\in \\GTM_{\\phi_\\gk}(\\Gw)$ and $\\gm\\in \\GTM(\\prt\\Gw)$ there exists a unique weak solution to problem $P_{\\gn,\\gm}$:"},"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":"1410.1201","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-05T19:42:27Z","cross_cats_sorted":[],"title_canon_sha256":"7f818bf3baa500d1930aee9c3d07264cd55269e6a867e23a31c83ac1623c1d17","abstract_canon_sha256":"86570cf72292fbd3dae811cfe433c1f2fbb0e2a8089f114d126671b23d6aca2a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:02.497780Z","signature_b64":"lkXIaBfGjXXVe417lWY4FjtGMOs29M0g+SSwgYZqX6UQj2LkGa8WqWavxGoTlQpi0Pk+Ruoe65hRJOFZrkrQAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ebd3090841711e75258a80b01dfe0975c65c4a71c0018a197784b72d9a1f548","last_reissued_at":"2026-05-18T02:41:02.497352Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:02.497352Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Measure boundary value problem for semilinear elliptic equations with critical Hardy potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Konstantinos Gkikas (CMM), Laurent Veron (LMPT)","submitted_at":"2014-10-05T19:42:27Z","abstract_excerpt":"Let $\\Omega\\subset\\BBR^N$ be a bounded $C^2$ domain and $\\CL_\\gk=-\\Gd-\\frac{\\gk}{d^2}$ the Hardy operator where $d=\\dist (.,\\prt\\Gw)$ and $0<\\gk\\leq\\frac{1}{4}$. Let $\\ga_{\\pm}=1\\pm\\sqrt{1-4\\gk}$ be the two Hardy exponents, $\\gl_\\gk$ the first eigenvalue of $\\CL_\\gk$ with corresponding positive eigenfunction $\\phi_\\gk$. If $g$ is a continuous nondecreasing function satisfying $\\int_1^\\infty(g(s)+|g(-s)|)s^{-2\\frac{2N-2+\\ga_+}{2N-4+\\ga_+}}ds<\\infty$, then for any Radon measures $\\gn\\in \\GTM_{\\phi_\\gk}(\\Gw)$ and $\\gm\\in \\GTM(\\prt\\Gw)$ there exists a unique weak solution to problem $P_{\\gn,\\gm}$:"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1201","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":"1410.1201","created_at":"2026-05-18T02:41:02.497414+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.1201v1","created_at":"2026-05-18T02:41:02.497414+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1201","created_at":"2026-05-18T02:41:02.497414+00:00"},{"alias_kind":"pith_short_12","alias_value":"L26TBEEEC4I6","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"L26TBEEEC4I6OUSY","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"L26TBEEE","created_at":"2026-05-18T12:28:35.611951+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/L26TBEEEC4I6OUSYVAFQDX7AS5","json":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5.json","graph_json":"https://pith.science/api/pith-number/L26TBEEEC4I6OUSYVAFQDX7AS5/graph.json","events_json":"https://pith.science/api/pith-number/L26TBEEEC4I6OUSYVAFQDX7AS5/events.json","paper":"https://pith.science/paper/L26TBEEE"},"agent_actions":{"view_html":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5","download_json":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5.json","view_paper":"https://pith.science/paper/L26TBEEE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.1201&json=true","fetch_graph":"https://pith.science/api/pith-number/L26TBEEEC4I6OUSYVAFQDX7AS5/graph.json","fetch_events":"https://pith.science/api/pith-number/L26TBEEEC4I6OUSYVAFQDX7AS5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5/action/storage_attestation","attest_author":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5/action/author_attestation","sign_citation":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5/action/citation_signature","submit_replication":"https://pith.science/pith/L26TBEEEC4I6OUSYVAFQDX7AS5/action/replication_record"}},"created_at":"2026-05-18T02:41:02.497414+00:00","updated_at":"2026-05-18T02:41:02.497414+00:00"}