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We prove that for any radial solution $u_p$ of \\eqref{problemAbstract} with $m$ nodal domains its Morse index $\\mathsf{m}(u_p)$ is given by the formula \\[\\mathsf{m}(u_p)=m+N(m-1)\\] if $p$ is sufficiently close to $p_S$."},"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":"1605.03357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-11T09:58:41Z","cross_cats_sorted":[],"title_canon_sha256":"4734a20e5e499ba7d6a7955eee7d830dd2a23ca36bf70499ab8ed6f49d5d6f42","abstract_canon_sha256":"fcc1eb776c6042c7f3220ce19d9692a41143753f41eca65054ba950f77fa6c50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:44.515503Z","signature_b64":"prqjRb2BHrn/6QiQxn/MD+xGmXIr+KthfLwkkS2sQaFAhfEtQBBj3GMOy0uChwfRcPPuojwJAyeZ1EbV+hGjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a82ffc5823d669cad3cb1a15705c48fbddd6be2d8608ae91fef7694dd3711ff8","last_reissued_at":"2026-05-18T01:01:44.514951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:44.514951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Morse index formula for radial solutions of Lane-Emden problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Filomena Pacella, Francesca De Marchis, Isabella Ianni","submitted_at":"2016-05-11T09:58:41Z","abstract_excerpt":"We consider the semilinear Lane-Emden problem: \\begin{equation}\\label{problemAbstract}\\left\\{\\begin{array}{lr}-\\Delta u= |u|^{p-1}u\\qquad \\mbox{ in }B u=0\\qquad\\qquad\\qquad\\mbox{ on }\\partial B \\end{array}\\right.\\tag{$\\mathcal E_p$} \\end{equation} where $B$ is the unit ball of $\\mathbb R^N$, $N\\geq3$, centered at the origin and $1<p<p_S$, $p_S=\\frac{N+2}{N-2}$. We prove that for any radial solution $u_p$ of \\eqref{problemAbstract} with $m$ nodal domains its Morse index $\\mathsf{m}(u_p)$ is given by the formula \\[\\mathsf{m}(u_p)=m+N(m-1)\\] if $p$ is sufficiently close to $p_S$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.03357","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":"1605.03357","created_at":"2026-05-18T01:01:44.515036+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.03357v1","created_at":"2026-05-18T01:01:44.515036+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.03357","created_at":"2026-05-18T01:01:44.515036+00:00"},{"alias_kind":"pith_short_12","alias_value":"VAX7YWBD2ZU4","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VAX7YWBD2ZU4VU6L","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VAX7YWBD","created_at":"2026-05-18T12:30:48.956258+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/VAX7YWBD2ZU4VU6LDIKXAXCI7P","json":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P.json","graph_json":"https://pith.science/api/pith-number/VAX7YWBD2ZU4VU6LDIKXAXCI7P/graph.json","events_json":"https://pith.science/api/pith-number/VAX7YWBD2ZU4VU6LDIKXAXCI7P/events.json","paper":"https://pith.science/paper/VAX7YWBD"},"agent_actions":{"view_html":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P","download_json":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P.json","view_paper":"https://pith.science/paper/VAX7YWBD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.03357&json=true","fetch_graph":"https://pith.science/api/pith-number/VAX7YWBD2ZU4VU6LDIKXAXCI7P/graph.json","fetch_events":"https://pith.science/api/pith-number/VAX7YWBD2ZU4VU6LDIKXAXCI7P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P/action/storage_attestation","attest_author":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P/action/author_attestation","sign_citation":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P/action/citation_signature","submit_replication":"https://pith.science/pith/VAX7YWBD2ZU4VU6LDIKXAXCI7P/action/replication_record"}},"created_at":"2026-05-18T01:01:44.515036+00:00","updated_at":"2026-05-18T01:01:44.515036+00:00"}