{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:F76KT6H42TPRY5CMBAVNLHS6YF","short_pith_number":"pith:F76KT6H4","schema_version":"1.0","canonical_sha256":"2ffca9f8fcd4df1c744c082ad59e5ec1658f61c2b624fc9dfd293c989c7e859c","source":{"kind":"arxiv","id":"1608.01742","version":2},"attestation_state":"computed","paper":{"title":"Multi-bump solutions for logarithmic Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chengxiang Zhang, Kazunaga Tanaka","submitted_at":"2016-08-05T02:32:12Z","abstract_excerpt":"We study spatially periodic logarithmic Schr\\\"odinger equations:\n  \\begin{equation}\\tag{LS}\n  -\\Delta u + V(x)u=Q(x)u\\log u^2, \\quad u>0\\quad \\text{in}\\ \\mathbb{R}^N,\n  \\end{equation} where $N\\geq 1$ and $V(x)$, $Q(x)$ are spatially $1$-periodic functions of class $C^1$. We take an approach using spatially $2L$-periodic problems ($L\\gg 1$) and we show the existence of infinitely many multi-bump solutions of $(LS)$ which are distinct under $\\mathbb{Z}^N$-action."},"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":"1608.01742","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-08-05T02:32:12Z","cross_cats_sorted":[],"title_canon_sha256":"9e545082e625665e37ef65aaedc7415e13542866657125620e1d5638e93db44e","abstract_canon_sha256":"b50e6166a3b4bb16eed99b042de615ff17a299b721c5a9b578f43d0687545ef6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:53.344414Z","signature_b64":"vWVWI6ThVuC54wCatVZ8suP/Kxh4RUNeg6UHDmHj7dFSz02DwABY85bxCOgrxCKL7mYd1M0Swi6T/1zF+w9nAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ffca9f8fcd4df1c744c082ad59e5ec1658f61c2b624fc9dfd293c989c7e859c","last_reissued_at":"2026-05-18T01:04:53.343906Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:53.343906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multi-bump solutions for logarithmic Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chengxiang Zhang, Kazunaga Tanaka","submitted_at":"2016-08-05T02:32:12Z","abstract_excerpt":"We study spatially periodic logarithmic Schr\\\"odinger equations:\n  \\begin{equation}\\tag{LS}\n  -\\Delta u + V(x)u=Q(x)u\\log u^2, \\quad u>0\\quad \\text{in}\\ \\mathbb{R}^N,\n  \\end{equation} where $N\\geq 1$ and $V(x)$, $Q(x)$ are spatially $1$-periodic functions of class $C^1$. We take an approach using spatially $2L$-periodic problems ($L\\gg 1$) and we show the existence of infinitely many multi-bump solutions of $(LS)$ which are distinct under $\\mathbb{Z}^N$-action."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01742","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":""},"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":"1608.01742","created_at":"2026-05-18T01:04:53.343976+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.01742v2","created_at":"2026-05-18T01:04:53.343976+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01742","created_at":"2026-05-18T01:04:53.343976+00:00"},{"alias_kind":"pith_short_12","alias_value":"F76KT6H42TPR","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"F76KT6H42TPRY5CM","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"F76KT6H4","created_at":"2026-05-18T12:30:15.759754+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/F76KT6H42TPRY5CMBAVNLHS6YF","json":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF.json","graph_json":"https://pith.science/api/pith-number/F76KT6H42TPRY5CMBAVNLHS6YF/graph.json","events_json":"https://pith.science/api/pith-number/F76KT6H42TPRY5CMBAVNLHS6YF/events.json","paper":"https://pith.science/paper/F76KT6H4"},"agent_actions":{"view_html":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF","download_json":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF.json","view_paper":"https://pith.science/paper/F76KT6H4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.01742&json=true","fetch_graph":"https://pith.science/api/pith-number/F76KT6H42TPRY5CMBAVNLHS6YF/graph.json","fetch_events":"https://pith.science/api/pith-number/F76KT6H42TPRY5CMBAVNLHS6YF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF/action/storage_attestation","attest_author":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF/action/author_attestation","sign_citation":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF/action/citation_signature","submit_replication":"https://pith.science/pith/F76KT6H42TPRY5CMBAVNLHS6YF/action/replication_record"}},"created_at":"2026-05-18T01:04:53.343976+00:00","updated_at":"2026-05-18T01:04:53.343976+00:00"}