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Our main result establishes that nonnegative solution $u\\in C^2(B_1\\setminus\\{0\\})$ of the above equation either has a removable singularity at the origin or behaves like \\begin{equation*} u(x) = A(1+o(1)) |x|^{-\\frac{2}{\\alpha-1}} \\left(\\log \\frac{1}{|x|}\\right)^{-\\frac{\\beta}{\\alpha-1}}\\quad\\text{as } x\\rightarrow 0, \\end{equation*} wi"},"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":"1804.04287","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-12T02:21:17Z","cross_cats_sorted":[],"title_canon_sha256":"df63dee056a061750dbb363e30f14b63917b9322fcec08ae849fae7f615aac3d","abstract_canon_sha256":"f8d2c0a3b4aff8c8ca0f8ec6b4d6e21bb5266ebd8ca8c49246614db7555f8700"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:40.042717Z","signature_b64":"mN/2lV6eXViFRZUkONMlcbhDO28lgHtRlQBkP+T58iZ200erqswMiwAo2K2zc+8K3ij148+Vr0sz+GRTKd3WBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cdfedd62d935a2e1f4398b2781809440874361a00caba5037e3d205e768af73","last_reissued_at":"2026-05-18T00:18:40.042000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:40.042000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Henrik Shahgholian, Marius Ghergu, Sunghan Kim","submitted_at":"2018-04-12T02:21:17Z","abstract_excerpt":"We study the semilinear elliptic equation \\begin{equation*} -\\Delta u=u^\\alpha |\\log u|^\\beta\\quad\\text{in }B_1\\setminus\\{0\\}, \\end{equation*} where $B_1\\subset\\mathbb{R}^n$ with $n\\geq 3$, $\\frac{n}{n-2} < \\alpha < \\frac{n+2}{n-2}$ and $-\\infty<\\beta<\\infty$. Our main result establishes that nonnegative solution $u\\in C^2(B_1\\setminus\\{0\\})$ of the above equation either has a removable singularity at the origin or behaves like \\begin{equation*} u(x) = A(1+o(1)) |x|^{-\\frac{2}{\\alpha-1}} \\left(\\log \\frac{1}{|x|}\\right)^{-\\frac{\\beta}{\\alpha-1}}\\quad\\text{as } x\\rightarrow 0, \\end{equation*} wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.04287","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":"1804.04287","created_at":"2026-05-18T00:18:40.042112+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.04287v1","created_at":"2026-05-18T00:18:40.042112+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.04287","created_at":"2026-05-18T00:18:40.042112+00:00"},{"alias_kind":"pith_short_12","alias_value":"PTP63VRNSNNC","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"PTP63VRNSNNC4H2D","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"PTP63VRN","created_at":"2026-05-18T12:32:46.962924+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/PTP63VRNSNNC4H2DTCZHQGAJIQ","json":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ.json","graph_json":"https://pith.science/api/pith-number/PTP63VRNSNNC4H2DTCZHQGAJIQ/graph.json","events_json":"https://pith.science/api/pith-number/PTP63VRNSNNC4H2DTCZHQGAJIQ/events.json","paper":"https://pith.science/paper/PTP63VRN"},"agent_actions":{"view_html":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ","download_json":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ.json","view_paper":"https://pith.science/paper/PTP63VRN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.04287&json=true","fetch_graph":"https://pith.science/api/pith-number/PTP63VRNSNNC4H2DTCZHQGAJIQ/graph.json","fetch_events":"https://pith.science/api/pith-number/PTP63VRNSNNC4H2DTCZHQGAJIQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ/action/storage_attestation","attest_author":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ/action/author_attestation","sign_citation":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ/action/citation_signature","submit_replication":"https://pith.science/pith/PTP63VRNSNNC4H2DTCZHQGAJIQ/action/replication_record"}},"created_at":"2026-05-18T00:18:40.042112+00:00","updated_at":"2026-05-18T00:18:40.042112+00:00"}