{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:VR5LXWVEI23CGOBTMJJ6VQ54FR","short_pith_number":"pith:VR5LXWVE","schema_version":"1.0","canonical_sha256":"ac7abbdaa446b62338336253eac3bc2c4d0410d4e09f6338d239fe7e641c42fa","source":{"kind":"arxiv","id":"2412.04794","version":2},"attestation_state":"computed","paper":{"title":"Multiplicity of solutions to a class of degenerate elliptic equations in both sub-critical and critical cases","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kaushik Bal, Sanjit Biswas","submitted_at":"2024-12-06T06:00:35Z","abstract_excerpt":"Given a smooth, bounded domain $\\Omega\\subset\\mathbb{R}^N$, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation \\begin{align*}\n  \\left. \\begin{array}{l}\n  -\\Delta_\\lambda u=\\mu g(z)|u|^{r-1}u+h(z)|u|^{s-1}u \\;\\text{in}\\; \\Omega\n  u\\in H^{1,\\lambda}_0(\\Omega)\n  \\end{array}\\right\\}\n  \\end{align*} where $\\Delta_\\lambda=\\Delta_x+|x|^{2\\lambda}\\Delta_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\\in\\Omega$, $N=n+m;\\, n,\\, m\\geq 1$, $\\lambda>0$, $0\\leq r<1<s<2^*_\\lambda-1$ and $\\mu$ is a positive parameter. The functions $g$ and $"},"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":"2412.04794","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-12-06T06:00:35Z","cross_cats_sorted":[],"title_canon_sha256":"7b9ea1a446bcc9b2a8c7fc90d008891e46dd30382ec4c470ee8307bb79bc3402","abstract_canon_sha256":"1c4917044ecc83a5d44ccddb51aa73c515bfad10dbd52ea04ce841b4e7533344"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:44.768910Z","signature_b64":"bTadnG7WPVe4BzXKZdJpCThnkV8ee/59xOOoGvNAS9195ajQTPnFLliWQTdWdHMHMQE0erpE+CzErKhNVQbACg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac7abbdaa446b62338336253eac3bc2c4d0410d4e09f6338d239fe7e641c42fa","last_reissued_at":"2026-06-19T16:12:44.768425Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:44.768425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicity of solutions to a class of degenerate elliptic equations in both sub-critical and critical cases","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kaushik Bal, Sanjit Biswas","submitted_at":"2024-12-06T06:00:35Z","abstract_excerpt":"Given a smooth, bounded domain $\\Omega\\subset\\mathbb{R}^N$, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation \\begin{align*}\n  \\left. \\begin{array}{l}\n  -\\Delta_\\lambda u=\\mu g(z)|u|^{r-1}u+h(z)|u|^{s-1}u \\;\\text{in}\\; \\Omega\n  u\\in H^{1,\\lambda}_0(\\Omega)\n  \\end{array}\\right\\}\n  \\end{align*} where $\\Delta_\\lambda=\\Delta_x+|x|^{2\\lambda}\\Delta_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\\in\\Omega$, $N=n+m;\\, n,\\, m\\geq 1$, $\\lambda>0$, $0\\leq r<1<s<2^*_\\lambda-1$ and $\\mu$ is a positive parameter. The functions $g$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.04794","kind":"arxiv","version":2},"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/2412.04794/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":"2412.04794","created_at":"2026-06-19T16:12:44.768487+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.04794v2","created_at":"2026-06-19T16:12:44.768487+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.04794","created_at":"2026-06-19T16:12:44.768487+00:00"},{"alias_kind":"pith_short_12","alias_value":"VR5LXWVEI23C","created_at":"2026-06-19T16:12:44.768487+00:00"},{"alias_kind":"pith_short_16","alias_value":"VR5LXWVEI23CGOBT","created_at":"2026-06-19T16:12:44.768487+00:00"},{"alias_kind":"pith_short_8","alias_value":"VR5LXWVE","created_at":"2026-06-19T16:12:44.768487+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.02020","citing_title":"Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity","ref_index":5,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR","json":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR.json","graph_json":"https://pith.science/api/pith-number/VR5LXWVEI23CGOBTMJJ6VQ54FR/graph.json","events_json":"https://pith.science/api/pith-number/VR5LXWVEI23CGOBTMJJ6VQ54FR/events.json","paper":"https://pith.science/paper/VR5LXWVE"},"agent_actions":{"view_html":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR","download_json":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR.json","view_paper":"https://pith.science/paper/VR5LXWVE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.04794&json=true","fetch_graph":"https://pith.science/api/pith-number/VR5LXWVEI23CGOBTMJJ6VQ54FR/graph.json","fetch_events":"https://pith.science/api/pith-number/VR5LXWVEI23CGOBTMJJ6VQ54FR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR/action/storage_attestation","attest_author":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR/action/author_attestation","sign_citation":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR/action/citation_signature","submit_replication":"https://pith.science/pith/VR5LXWVEI23CGOBTMJJ6VQ54FR/action/replication_record"}},"created_at":"2026-06-19T16:12:44.768487+00:00","updated_at":"2026-06-19T16:12:44.768487+00:00"}