{"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"}