{"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"}