{"paper":{"title":"Decay rate and radial symmetry of the exponential elliptic equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui, SungHoon Kim","submitted_at":"2012-09-04T07:15:36Z","abstract_excerpt":"Let $n\\geq 3$, $\\alpha$, $\\beta\\in\\mathbb{R}$, and let $v$ be a solution $\\Delta v+\\alpha e^v+\\beta x\\cdot\\nabla e^v=0$ in $\\mathbb{R}^n$, which satisfies the conditions $\\lim_{R\\to\\infty}\\frac{1}{\\log R}\\int_{1}^{R}\\rho^{1-n} (\\int_{B_{\\rho}}e^v\\,dx)d\\rho\\in (0,\\infty)$ and $|x|^2e^{v(x)}\\le A_1$ in $\\R^n$. We prove that $\\frac{v(x)}{\\log |x|}\\to -2$ as $|x|\\to\\infty$ and $\\alpha>2\\beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0544","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"}