{"paper":{"title":"On a power-type coupled system of Monge-Amp\\`{e}re equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Zexin Qi, Zhitao Zhang","submitted_at":"2014-12-11T02:18:47Z","abstract_excerpt":"We study an elliptic system coupled by Monge-Amp\\`{e}re equations:\n  \\begin{center}\n  $\\left\\{\n  \\begin{array}{ll}\n  det~D^{2}u_{1}={(-u_{2})}^\\alpha, & \\hbox{in $\\Omega,$}\n  det~D^{2}u_{2}={(-u_{1})}^\\beta, & \\hbox{in $\\Omega,$}\n  u_{1}<0, u_{2}<0,& \\hbox{in $\\Omega,$}\n  u_{1}=u_{2}=0, & \\hbox{on $ \\partial \\Omega,$}\n  \\end{array}\n  \\right.$\n  \\end{center} here $\\Omega$~is a smooth, bounded and strictly convex domain in~$\\mathbb{R}^{N}$,~$N\\geq2,~\\alpha >0,~\\beta >0$. When $\\Omega$ is the unit ball in $\\mathbb{R}^{N}$, we use index theory of fixed points for completely continuous operators to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.3519","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"}