{"paper":{"title":"Existence of entire solutions of Monge-Amp\\`ere equations with prescribed asymptotic behaviors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiguang Bao, Jingang Xiong, Ziwei Zhou","submitted_at":"2018-09-14T13:47:39Z","abstract_excerpt":"We prove the existence of entire solutions of the Monge-Amp\\`ere equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due to the global logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Amp\\`ere equation $\\det \\nabla^2 u=1$ with $k\\ge 2$ singular points, which was established by G\\'alvez-Mart\\'inez-Mira in 2005. We also obtain the existence in higher dimensional "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05421","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"}