{"paper":{"title":"Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaobao Zhu, Yunyan Yang","submitted_at":"2017-06-26T02:20:21Z","abstract_excerpt":"Let $(\\Sigma,g)$ be a compact Riemannian surface without boundary and $\\lambda_1(\\Sigma)$ be the first eigenvalue of the Laplace-Beltrami operator $\\Delta_g$. Let $h$ be a positive smooth function on $\\Sigma$. Define a functional $$J_{\\alpha,\\beta}(u)=\\frac{1}{2}\\int_\\Sigma(|\\nabla_gu|^2-\\alpha u^2)dv_g-\\beta\\log\\int_\\Sigma he^udv_g$$ on a function space $\\mathcal{H}=\\left\\{u\\in W^{1,2}(\\Sigma): \\int_\\Sigma udv_g=0\\right\\}$. If $\\alpha<\\lambda_1(\\Sigma)$ and $J_{\\alpha,8\\pi}$ has no minimizer on $\\mathcal{H}$, then we calculate the infimum of $J_{\\alpha,8\\pi}$ on $\\mathcal{H}$ by using the met"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08207","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"}