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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","authors_text":"Hardy Chan, Luiz Fernando de Oliveira Faria, Nassif Ghoussoub, Saikat Mazumdar, Shaya Shakerian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-10-03T16:43:16Z","title":"Mass and Extremals Associated with the Hardy-Schr\\\"odinger Operator on Hyperbolic Space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01271","kind":"arxiv","version":2},"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":"sha256:93bfa3a4b11b341d9494e41dfc04c6c94b6134a40e383d5e6a8a2dcf21143962","target":"record","created_at":"2026-05-18T00:19:08Z","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":"6e463f1740c146c4255ac8b0653f8cc318897c322184539e5123d881a7533344","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-10-03T16:43:16Z","title_canon_sha256":"5aec6fabddd6148cd3c1074077a0a6a7b7ccac21b302999bbdfce476b98a2a48"},"schema_version":"1.0","source":{"id":"1710.01271","kind":"arxiv","version":2}},"canonical_sha256":"fc44a250c77fad451bc0560940a8f7ac6d9e9d935807cff0e91315c15bff1335","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc44a250c77fad451bc0560940a8f7ac6d9e9d935807cff0e91315c15bff1335","first_computed_at":"2026-05-18T00:19:08.711000Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:08.711000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QgPPHrvGeXoMRjlxwwYsFCTs1WgJCcVQe6Co+qJAuGJihsnFxJ/keNadl1biGd3Y86B5bfQIHq/HeQiePvILDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:08.711702Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.01271","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93bfa3a4b11b341d9494e41dfc04c6c94b6134a40e383d5e6a8a2dcf21143962","sha256:b7b64dfe331104ee78033f79db8332c3e465dde77e5392083a41b283d834661c"],"state_sha256":"157338553496ce6ac43fadbc814e5b55bd99d4cf2e7350b55f6da5b23ca3ae75"}