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Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\\sim \\frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -\\Delta_{\\mathbb{B}^n}-\\gamma{V_2}$ is positive definite whenever $\\gamma <\\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-\\Delta_{\\mathbb{B}^n}u-\\gamma{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\\qu"},"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":"1710.01271","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-10-03T16:43:16Z","cross_cats_sorted":[],"title_canon_sha256":"5aec6fabddd6148cd3c1074077a0a6a7b7ccac21b302999bbdfce476b98a2a48","abstract_canon_sha256":"6e463f1740c146c4255ac8b0653f8cc318897c322184539e5123d881a7533344"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:08.711702Z","signature_b64":"QgPPHrvGeXoMRjlxwwYsFCTs1WgJCcVQe6Co+qJAuGJihsnFxJ/keNadl1biGd3Y86B5bfQIHq/HeQiePvILDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc44a250c77fad451bc0560940a8f7ac6d9e9d935807cff0e91315c15bff1335","last_reissued_at":"2026-05-18T00:19:08.711000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:08.711000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mass and Extremals Associated with the Hardy-Schr\\\"odinger Operator on Hyperbolic Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hardy Chan, Luiz Fernando de Oliveira Faria, Nassif Ghoussoub, Saikat Mazumdar, Shaya Shakerian","submitted_at":"2017-10-03T16:43:16Z","abstract_excerpt":"We consider the Hardy-Schr\\\"odinger operator $ -\\Delta_{\\mathbb{B}^n}-\\gamma{V_2}$ on the Poincar\\'e ball model of the Hyperbolic space ${\\mathbb{B}^n}$ ($n \\geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\\sim \\frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -\\Delta_{\\mathbb{B}^n}-\\gamma{V_2}$ is positive definite whenever $\\gamma <\\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-\\Delta_{\\mathbb{B}^n}u-\\gamma{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\\qu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01271","kind":"arxiv","version":2},"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":"1710.01271","created_at":"2026-05-18T00:19:08.711118+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.01271v2","created_at":"2026-05-18T00:19:08.711118+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.01271","created_at":"2026-05-18T00:19:08.711118+00:00"},{"alias_kind":"pith_short_12","alias_value":"7RCKEUGHP6WU","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"7RCKEUGHP6WUKG6A","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"7RCKEUGH","created_at":"2026-05-18T12:31:05.417338+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/7RCKEUGHP6WUKG6AKYEUBKHXVR","json":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR.json","graph_json":"https://pith.science/api/pith-number/7RCKEUGHP6WUKG6AKYEUBKHXVR/graph.json","events_json":"https://pith.science/api/pith-number/7RCKEUGHP6WUKG6AKYEUBKHXVR/events.json","paper":"https://pith.science/paper/7RCKEUGH"},"agent_actions":{"view_html":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR","download_json":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR.json","view_paper":"https://pith.science/paper/7RCKEUGH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.01271&json=true","fetch_graph":"https://pith.science/api/pith-number/7RCKEUGHP6WUKG6AKYEUBKHXVR/graph.json","fetch_events":"https://pith.science/api/pith-number/7RCKEUGHP6WUKG6AKYEUBKHXVR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR/action/storage_attestation","attest_author":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR/action/author_attestation","sign_citation":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR/action/citation_signature","submit_replication":"https://pith.science/pith/7RCKEUGHP6WUKG6AKYEUBKHXVR/action/replication_record"}},"created_at":"2026-05-18T00:19:08.711118+00:00","updated_at":"2026-05-18T00:19:08.711118+00:00"}