{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:GJKPRWPP54MGWD2JCPKRR2R7C6","short_pith_number":"pith:GJKPRWPP","schema_version":"1.0","canonical_sha256":"3254f8d9efef186b0f4913d518ea3f17812c42a1241ac2e315a9d7c2be4d314e","source":{"kind":"arxiv","id":"1909.00492","version":2},"attestation_state":"computed","paper":{"title":"Classification of nonnegative solutions to static Schr\\\"{o}dinger-Hartree-Maxwell type equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guolin Qin, Wei Dai, Zhao Liu","submitted_at":"2019-09-01T23:45:37Z","abstract_excerpt":"In this paper, we are mainly concerned with the physically interesting static Schr\\\"{o}dinger-Hartree-Maxwell type equations \\begin{equation*}\n  (-\\Delta)^{s}u(x)=\\left(\\frac{1}{|x|^{\\sigma}}\\ast |u|^{p}\\right)u^{q}(x) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\text{in} \\,\\,\\, \\mathbb{R}^{n} \\end{equation*} involving higher-order or higher-order fractional Laplacians, where $n\\geq1$, $0<s:=m+\\frac{\\alpha}{2}<\\frac{n}{2}$, $m\\geq0$ is an integer, $0<\\alpha\\leq2$, $0<\\sigma<n$, $0<p\\leq\\frac{2n-\\sigma}{n-2s}$ and $0<q\\leq\\frac{n+2s-\\sigma}{n-2s}$. We first prove the super poly-harmonic properties of nonnegative "},"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":"1909.00492","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-09-01T23:45:37Z","cross_cats_sorted":[],"title_canon_sha256":"aa82229a7154247feaaabeddaec381cfb7628ecc8de5e2924fa568a97294a819","abstract_canon_sha256":"1361045b3a99ca41d4521ad9ed0dc00be27b37114776e7b826389018ae9c1270"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:22:34.152238Z","signature_b64":"1K6SQu9Sib0N33WlThxhqJI0MobV9qdyxWJsHeqvou0F0Ab0jH8NHnE8+7Pi5DPO+bsnqaDBNkDIyQofr4WDDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3254f8d9efef186b0f4913d518ea3f17812c42a1241ac2e315a9d7c2be4d314e","last_reissued_at":"2026-07-05T02:22:34.151801Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:22:34.151801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of nonnegative solutions to static Schr\\\"{o}dinger-Hartree-Maxwell type equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guolin Qin, Wei Dai, Zhao Liu","submitted_at":"2019-09-01T23:45:37Z","abstract_excerpt":"In this paper, we are mainly concerned with the physically interesting static Schr\\\"{o}dinger-Hartree-Maxwell type equations \\begin{equation*}\n  (-\\Delta)^{s}u(x)=\\left(\\frac{1}{|x|^{\\sigma}}\\ast |u|^{p}\\right)u^{q}(x) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\text{in} \\,\\,\\, \\mathbb{R}^{n} \\end{equation*} involving higher-order or higher-order fractional Laplacians, where $n\\geq1$, $0<s:=m+\\frac{\\alpha}{2}<\\frac{n}{2}$, $m\\geq0$ is an integer, $0<\\alpha\\leq2$, $0<\\sigma<n$, $0<p\\leq\\frac{2n-\\sigma}{n-2s}$ and $0<q\\leq\\frac{n+2s-\\sigma}{n-2s}$. We first prove the super poly-harmonic properties of nonnegative "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.00492","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/1909.00492/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":"1909.00492","created_at":"2026-07-05T02:22:34.151854+00:00"},{"alias_kind":"arxiv_version","alias_value":"1909.00492v2","created_at":"2026-07-05T02:22:34.151854+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.00492","created_at":"2026-07-05T02:22:34.151854+00:00"},{"alias_kind":"pith_short_12","alias_value":"GJKPRWPP54MG","created_at":"2026-07-05T02:22:34.151854+00:00"},{"alias_kind":"pith_short_16","alias_value":"GJKPRWPP54MGWD2J","created_at":"2026-07-05T02:22:34.151854+00:00"},{"alias_kind":"pith_short_8","alias_value":"GJKPRWPP","created_at":"2026-07-05T02:22:34.151854+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/GJKPRWPP54MGWD2JCPKRR2R7C6","json":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6.json","graph_json":"https://pith.science/api/pith-number/GJKPRWPP54MGWD2JCPKRR2R7C6/graph.json","events_json":"https://pith.science/api/pith-number/GJKPRWPP54MGWD2JCPKRR2R7C6/events.json","paper":"https://pith.science/paper/GJKPRWPP"},"agent_actions":{"view_html":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6","download_json":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6.json","view_paper":"https://pith.science/paper/GJKPRWPP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1909.00492&json=true","fetch_graph":"https://pith.science/api/pith-number/GJKPRWPP54MGWD2JCPKRR2R7C6/graph.json","fetch_events":"https://pith.science/api/pith-number/GJKPRWPP54MGWD2JCPKRR2R7C6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6/action/storage_attestation","attest_author":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6/action/author_attestation","sign_citation":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6/action/citation_signature","submit_replication":"https://pith.science/pith/GJKPRWPP54MGWD2JCPKRR2R7C6/action/replication_record"}},"created_at":"2026-07-05T02:22:34.151854+00:00","updated_at":"2026-07-05T02:22:34.151854+00:00"}