{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7LL5QXH4J3C3OIK5O5IAPWMFNG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0a66ae0bbc9a7022ffea462089824bb35ebe6df7cc5d1420062291ab18ab4589","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-02T14:50:42Z","title_canon_sha256":"126b3b47dc0233a2c5a1a62bdb73910237ba807607b1dcc3a8398f25f934d7d8"},"schema_version":"1.0","source":{"id":"1308.0519","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.0519","created_at":"2026-05-18T02:20:20Z"},{"alias_kind":"arxiv_version","alias_value":"1308.0519v3","created_at":"2026-05-18T02:20:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.0519","created_at":"2026-05-18T02:20:20Z"},{"alias_kind":"pith_short_12","alias_value":"7LL5QXH4J3C3","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7LL5QXH4J3C3OIK5","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7LL5QXH4","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:f8a363cfb448fcc42f5ca3478cd9cac324b17104a3f1f425e68e6a06558f255e","target":"graph","created_at":"2026-05-18T02:20:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we study the problem\n  -\\Delta u =\\left(\\frac{2+\\alpha}{2}\\right)^2\\abs{x}^{\\alpha}f(\\lambda,u), & \\hbox{in}B_1 \\\\ u > 0, & \\hbox{in}B_1 u = 0, & \\hbox{on} \\partial B_1 where $B_1$ is the unit ball of $\\R^2$, $f$ is a smooth nonlinearity and $\\a$, $\\l$ are real numbers with $\\a>0$. From a careful study of the linearized operator we compute the Morse index of some radial solutions to \\eqref{i0}. Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter $\\l$. The case $f(\\lambda,u)=\\l e^u$ provide","authors_text":"Francesca Gladiali, Massimo Grossi, S\\'ergio Neves","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-02T14:50:42Z","title":"Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0519","kind":"arxiv","version":3},"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":"sha256:829abef9922131e9ee7208d802bbc0ce6e035aa9d678bea5dcb4de9c526327aa","target":"record","created_at":"2026-05-18T02:20:20Z","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":"0a66ae0bbc9a7022ffea462089824bb35ebe6df7cc5d1420062291ab18ab4589","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-02T14:50:42Z","title_canon_sha256":"126b3b47dc0233a2c5a1a62bdb73910237ba807607b1dcc3a8398f25f934d7d8"},"schema_version":"1.0","source":{"id":"1308.0519","kind":"arxiv","version":3}},"canonical_sha256":"fad7d85cfc4ec5b7215d775007d98569b575b1508b2e701b88aa9b8ff8994c67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fad7d85cfc4ec5b7215d775007d98569b575b1508b2e701b88aa9b8ff8994c67","first_computed_at":"2026-05-18T02:20:20.108441Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:20:20.108441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aId4vCHWR3IW7k9VtNh7x428FFvVplIkg6+ke0Hg9EZozBh57bEjLwYQ/bi8oJN+DVSte+8NwialNlvSDpu8CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:20:20.109113Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.0519","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:829abef9922131e9ee7208d802bbc0ce6e035aa9d678bea5dcb4de9c526327aa","sha256:f8a363cfb448fcc42f5ca3478cd9cac324b17104a3f1f425e68e6a06558f255e"],"state_sha256":"2e6d5d74aec5c9afe68e086a84d0c9e72b65b734faca11a0211740dd2670def4"}