{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:KCRSZ2UOINFPMWG72YJHLPJOPW","short_pith_number":"pith:KCRSZ2UO","schema_version":"1.0","canonical_sha256":"50a32cea8e434af658dfd61275bd2e7daf0ac24ade9b45ba0a7c11477d234e5c","source":{"kind":"arxiv","id":"1509.05839","version":1},"attestation_state":"computed","paper":{"title":"Weak solutions of semilinear elliptic equation involving Dirac mass","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huyuan Chen, Jianfu Yang, Patricio Felmer","submitted_at":"2015-09-19T01:31:48Z","abstract_excerpt":"In this paper, we study the following elliptic problem with Dirac mass \\begin{equation}\\label{eq 0.1}\n  -\\Delta u=Vu^p+k \\delta_0\\quad\n  {\\rm in}\\quad \\mathbb{R}^N, \\qquad \\lim_{|x|\\to+\\infty}u(x)=0,\n  \\end{equation} where $N>2$, $p>0$, $k>0$, $\\delta_0$ is Dirac mass at the origin, the function $V$ is a locally Lipchitz continuous in $\\mathbb{R}^N\\setminus\\{0\\}$ satisfying $$ V(x)\\le \\frac{c_1}{|x|^{a_0}(1+|x|^{a_\\infty-a_0})} $$ with $a_0<N,\\ a_\\infty>a_0 $ and $c_1>0$. We obtain two positive solutions of (\\ref{eq 0.1}) with additional conditions for parameters on $a_\\infty, a_0$, $p$ and $k"},"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":"1509.05839","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2015-09-19T01:31:48Z","cross_cats_sorted":[],"title_canon_sha256":"02f3e12aece3e65cf7f1f9ecd82721ff171e982ad31edcd2396332d3a6c84088","abstract_canon_sha256":"9c8a7a5ca3266488c1054c8dbde6977574476ef3070e68f39114a1ea66314f67"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:36.555363Z","signature_b64":"p21cYyjXlfHLefah1b3r6TuhCxInSMOdV3NZnFU/4bgpux8HClxFkoYofXF4cMI+etyOQeaFNTOVN1QQjraXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50a32cea8e434af658dfd61275bd2e7daf0ac24ade9b45ba0a7c11477d234e5c","last_reissued_at":"2026-05-18T01:32:36.554922Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:36.554922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak solutions of semilinear elliptic equation involving Dirac mass","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huyuan Chen, Jianfu Yang, Patricio Felmer","submitted_at":"2015-09-19T01:31:48Z","abstract_excerpt":"In this paper, we study the following elliptic problem with Dirac mass \\begin{equation}\\label{eq 0.1}\n  -\\Delta u=Vu^p+k \\delta_0\\quad\n  {\\rm in}\\quad \\mathbb{R}^N, \\qquad \\lim_{|x|\\to+\\infty}u(x)=0,\n  \\end{equation} where $N>2$, $p>0$, $k>0$, $\\delta_0$ is Dirac mass at the origin, the function $V$ is a locally Lipchitz continuous in $\\mathbb{R}^N\\setminus\\{0\\}$ satisfying $$ V(x)\\le \\frac{c_1}{|x|^{a_0}(1+|x|^{a_\\infty-a_0})} $$ with $a_0<N,\\ a_\\infty>a_0 $ and $c_1>0$. We obtain two positive solutions of (\\ref{eq 0.1}) with additional conditions for parameters on $a_\\infty, a_0$, $p$ and $k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05839","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":"1509.05839","created_at":"2026-05-18T01:32:36.554990+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05839v1","created_at":"2026-05-18T01:32:36.554990+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05839","created_at":"2026-05-18T01:32:36.554990+00:00"},{"alias_kind":"pith_short_12","alias_value":"KCRSZ2UOINFP","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"KCRSZ2UOINFPMWG7","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"KCRSZ2UO","created_at":"2026-05-18T12:29:27.538025+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/KCRSZ2UOINFPMWG72YJHLPJOPW","json":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW.json","graph_json":"https://pith.science/api/pith-number/KCRSZ2UOINFPMWG72YJHLPJOPW/graph.json","events_json":"https://pith.science/api/pith-number/KCRSZ2UOINFPMWG72YJHLPJOPW/events.json","paper":"https://pith.science/paper/KCRSZ2UO"},"agent_actions":{"view_html":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW","download_json":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW.json","view_paper":"https://pith.science/paper/KCRSZ2UO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05839&json=true","fetch_graph":"https://pith.science/api/pith-number/KCRSZ2UOINFPMWG72YJHLPJOPW/graph.json","fetch_events":"https://pith.science/api/pith-number/KCRSZ2UOINFPMWG72YJHLPJOPW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW/action/storage_attestation","attest_author":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW/action/author_attestation","sign_citation":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW/action/citation_signature","submit_replication":"https://pith.science/pith/KCRSZ2UOINFPMWG72YJHLPJOPW/action/replication_record"}},"created_at":"2026-05-18T01:32:36.554990+00:00","updated_at":"2026-05-18T01:32:36.554990+00:00"}