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When $k=\\frac4n$, problem $(E)$ only has the solutions with the form $$u_{\\tilde x,t}(x)=\\beta_n \\Big(\\frac{t}{t^2+|x-\\tilde x|^2)}\\Big)^{\\frac{n}{2}}\\quad \\text{ for any $t>0$, $\\tilde x\\in"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2409.04797","kind":"arxiv","version":6},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-09-07T11:27:28Z","cross_cats_sorted":[],"title_canon_sha256":"2e4e91a8ed0b1720f21eaf34cc1e63f25d1eaa78cfc1007f71e82e802518ca8e","abstract_canon_sha256":"343b3e75bdf3c1e9362921f62acedf28fceefb94769a50ff60b7097a3cc4f489"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:35:50.902097Z","signature_b64":"9Xkr0otJRuBrW/IDbfqAbZ7rZpYJGVv/h/xC2YVNsh5a4VV76SQqKyN5OQUqEjMRaI9AmAcHtc4GaMm56MUqCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abe8667187e21cacb4ca73124a973f0723996b03a5757fd2c200f35d0b536f6e","last_reissued_at":"2026-07-05T11:35:50.901674Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:35:50.901674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On positive solutions of critical semilinear equations involving the Logarithmic Laplacian","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Feng Zhou, Huyuan Chen","submitted_at":"2024-09-07T11:27:28Z","abstract_excerpt":"In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E)\\qquad \\qquad\\qquad\\qquad\\qquad \\mathcal{L}_\\Delta u= k u\\log u,\\qquad u\\geq0 \\quad \\ {\\rm in}\\ \\ \\mathbb{R}^n, \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad$$ where $k\\in(0,+\\infty)$, $\\mathcal{L}_\\Delta$ is the logarithmic Laplacian in $\\mathbb{R}^n$ with $n\\in\\mathbb{N}$, and $s\\log s=0$ if $s=0$. When $k=\\frac4n$, problem $(E)$ only has the solutions with the form $$u_{\\tilde x,t}(x)=\\beta_n \\Big(\\frac{t}{t^2+|x-\\tilde x|^2)}\\Big)^{\\frac{n}{2}}\\quad \\text{ for any $t>0$, $\\tilde x\\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.04797","kind":"arxiv","version":6},"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/2409.04797/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":"2409.04797","created_at":"2026-07-05T11:35:50.901730+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.04797v6","created_at":"2026-07-05T11:35:50.901730+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.04797","created_at":"2026-07-05T11:35:50.901730+00:00"},{"alias_kind":"pith_short_12","alias_value":"VPUGM4MH4IOK","created_at":"2026-07-05T11:35:50.901730+00:00"},{"alias_kind":"pith_short_16","alias_value":"VPUGM4MH4IOKZNGK","created_at":"2026-07-05T11:35:50.901730+00:00"},{"alias_kind":"pith_short_8","alias_value":"VPUGM4MH","created_at":"2026-07-05T11:35:50.901730+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.21779","citing_title":"The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces","ref_index":16,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4","json":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4.json","graph_json":"https://pith.science/api/pith-number/VPUGM4MH4IOKZNGKOMJEVFZ7A4/graph.json","events_json":"https://pith.science/api/pith-number/VPUGM4MH4IOKZNGKOMJEVFZ7A4/events.json","paper":"https://pith.science/paper/VPUGM4MH"},"agent_actions":{"view_html":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4","download_json":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4.json","view_paper":"https://pith.science/paper/VPUGM4MH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.04797&json=true","fetch_graph":"https://pith.science/api/pith-number/VPUGM4MH4IOKZNGKOMJEVFZ7A4/graph.json","fetch_events":"https://pith.science/api/pith-number/VPUGM4MH4IOKZNGKOMJEVFZ7A4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4/action/storage_attestation","attest_author":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4/action/author_attestation","sign_citation":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4/action/citation_signature","submit_replication":"https://pith.science/pith/VPUGM4MH4IOKZNGKOMJEVFZ7A4/action/replication_record"}},"created_at":"2026-07-05T11:35:50.901730+00:00","updated_at":"2026-07-05T11:35:50.901730+00:00"}