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In this setting, the problem can be extended to consider real values of $n.$ We show that if $2<n<4$ this problem has a unique positive solution if and only if $\\lambda\\in \\left(n(n-2)/4 +L^*\\,,\\, \\lambda_1\\right).$ Here $L^*$ is the first positive value of $L = -\\ell"},"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":"1507.05318","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-19T18:45:17Z","cross_cats_sorted":[],"title_canon_sha256":"b9f831786e37dc2b932435da914a870f68ea12e64ab9274c58f46e0ea70ea28d","abstract_canon_sha256":"74e89ab7bd80d0ee833faf72d5d3eae6401af7a090d0817763055854bd700787"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:37.753410Z","signature_b64":"XJcJr8PXOUzzGfWm2RxFxLdq4c78b2vx8IcUfBBNnsko/AvRTQNPSGfkE5ZxoOh8W2+dcJfsErHcWqOXLrTvDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6da9eb5ac47f96f59c8ede487d27e07a5bcf9fef1a9b1bd353b69112c3ffc49","last_reissued_at":"2026-05-18T01:22:37.752898Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:37.752898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The solution gap of the Brezis-Nirenberg problem on the hyperbolic space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Soledad Benguria","submitted_at":"2015-07-19T18:45:17Z","abstract_excerpt":"We consider the positive solutions of the nonlinear eigenvalue problem $-\\Delta_{\\mathbb{H}^n} u = \\lambda u + u^p, $ with $p=\\frac{n+2}{n-2}$ and $u \\in H_0^1(\\Omega),$ where $\\Omega$ is a geodesic ball of radius $\\theta_1$ on $\\mathbb{H}^n.$ For radial solutions, this equation can be written as an ODE having $n$ as a parameter. 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