{"paper":{"title":"On Gaussian curvature equations in $\\mathbb{R}^2$ with prescribed non-positive curvature","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dong Ye, Feng Zhou, Huyuan Chen","submitted_at":"2018-10-17T03:16:33Z","abstract_excerpt":"The purpose of this paper is to study the solutions of $$ \\Delta u +K(x) e^{2u}=0 \\quad{\\rm in}\\;\\; \\mathbb{R}^2 $$ with $K\\le 0$. We introduce the following quantity: $$\\alpha_p(K)=\\sup\\left\\{\\alpha \\in \\mathbb{R}:\\, \\int_{\\mathbb{R}^2} |K(x)|^p(1+|x|)^{2\\alpha p+2(p-1)} dx<+\\infty\\right\\}, \\quad \\forall\\; p \\ge 1.$$ Under the assumption $({\\mathbb H}_1)$: $\\alpha_p(K)> -\\infty$ for some $p>1$ and $\\alpha_1(K) > 0$, we show that for any $0 < \\alpha < \\alpha_1(K)$, there is a unique solution $u_\\alpha$ with $u_\\alpha(x) = \\alpha \\ln |x|+ c_\\alpha+o\\big(|x|^{-\\frac{2\\beta}{1+2\\beta}} \\big)$ at "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07369","kind":"arxiv","version":2},"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"}