{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:ZIVI52HOUWN5QCTKIDIGELNPAX","short_pith_number":"pith:ZIVI52HO","schema_version":"1.0","canonical_sha256":"ca2a8ee8eea59bd80a6a40d0622daf05fbbd489fe08481993c9cdc3814673cfb","source":{"kind":"arxiv","id":"2007.10838","version":1},"attestation_state":"computed","paper":{"title":"Solutions of super-linear elliptic equations and their Morse indices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Foued Mtiri","submitted_at":"2020-07-20T07:45:17Z","abstract_excerpt":"We investigate here the degenerate bi-harmonic equation: $$\\Delta_{m}^2 u=f(x,u)\\; \\;\\;\\mbox{in} \\O,\\quad u = \\Delta u = 0\\quad \\mbox{on }\\; \\p\\Omega,$$ with $m\\ge 2,$ and also the degenerate tri-harmonic equation: $$ -\\Delta_{m}^3 u=f(x,u)\\;\\;\\; \\mbox{in} \\O,\\quad u = \\frac{\\p u}{\\p \\nu} = \\frac{\\p^{2} u}{\\p\\nu^{2}} = 0\\quad \\mbox{on }\\; \\p\\Omega,$$ where $\\Omega\\subset \\mathbb{R}^{N}$ is a bounded domain with smooth boundary $N>4$ or $N>6$ resp, and $f \\in \\mathrm{C}^{1}(\\Omega\\times \\mathbb{R})$ satisfying suitable m-superlinear and subcritical growth conditions. Our main purpose is to esta"},"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":"2007.10838","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-07-20T07:45:17Z","cross_cats_sorted":[],"title_canon_sha256":"e26dece989e6de88b9405f9793f5a752c62847dc920901213992d34d3cad6c0b","abstract_canon_sha256":"bf19e42b94a525474fcd31c5604dc46994a782177b39decb1c5424d17fbc52b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:21:12.510860Z","signature_b64":"uIcC/hpIepwklaMAxYQ042TSLHMyLv89R9980fpmM2SVbikKp0ezv4I4qGn1tCoA7wxjRYHnF2CXK+djVSbaCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca2a8ee8eea59bd80a6a40d0622daf05fbbd489fe08481993c9cdc3814673cfb","last_reissued_at":"2026-07-05T01:21:12.510434Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:21:12.510434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Solutions of super-linear elliptic equations and their Morse indices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Foued Mtiri","submitted_at":"2020-07-20T07:45:17Z","abstract_excerpt":"We investigate here the degenerate bi-harmonic equation: $$\\Delta_{m}^2 u=f(x,u)\\; \\;\\;\\mbox{in} \\O,\\quad u = \\Delta u = 0\\quad \\mbox{on }\\; \\p\\Omega,$$ with $m\\ge 2,$ and also the degenerate tri-harmonic equation: $$ -\\Delta_{m}^3 u=f(x,u)\\;\\;\\; \\mbox{in} \\O,\\quad u = \\frac{\\p u}{\\p \\nu} = \\frac{\\p^{2} u}{\\p\\nu^{2}} = 0\\quad \\mbox{on }\\; \\p\\Omega,$$ where $\\Omega\\subset \\mathbb{R}^{N}$ is a bounded domain with smooth boundary $N>4$ or $N>6$ resp, and $f \\in \\mathrm{C}^{1}(\\Omega\\times \\mathbb{R})$ satisfying suitable m-superlinear and subcritical growth conditions. Our main purpose is to esta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.10838","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2007.10838/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2007.10838","created_at":"2026-07-05T01:21:12.510497+00:00"},{"alias_kind":"arxiv_version","alias_value":"2007.10838v1","created_at":"2026-07-05T01:21:12.510497+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.10838","created_at":"2026-07-05T01:21:12.510497+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZIVI52HOUWN5","created_at":"2026-07-05T01:21:12.510497+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZIVI52HOUWN5QCTK","created_at":"2026-07-05T01:21:12.510497+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZIVI52HO","created_at":"2026-07-05T01:21:12.510497+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/ZIVI52HOUWN5QCTKIDIGELNPAX","json":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX.json","graph_json":"https://pith.science/api/pith-number/ZIVI52HOUWN5QCTKIDIGELNPAX/graph.json","events_json":"https://pith.science/api/pith-number/ZIVI52HOUWN5QCTKIDIGELNPAX/events.json","paper":"https://pith.science/paper/ZIVI52HO"},"agent_actions":{"view_html":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX","download_json":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX.json","view_paper":"https://pith.science/paper/ZIVI52HO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2007.10838&json=true","fetch_graph":"https://pith.science/api/pith-number/ZIVI52HOUWN5QCTKIDIGELNPAX/graph.json","fetch_events":"https://pith.science/api/pith-number/ZIVI52HOUWN5QCTKIDIGELNPAX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX/action/storage_attestation","attest_author":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX/action/author_attestation","sign_citation":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX/action/citation_signature","submit_replication":"https://pith.science/pith/ZIVI52HOUWN5QCTKIDIGELNPAX/action/replication_record"}},"created_at":"2026-07-05T01:21:12.510497+00:00","updated_at":"2026-07-05T01:21:12.510497+00:00"}