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Existence of entire solutions of Monge–Ampère equations with prescribed asymptotic behavior

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Abstract

We prove the existence of entire solutions of the Monge–Ampère equations with prescribed asymptotic behavior at infinity of the plane, which was left unsolved 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–Ampère equation \(\det \nabla ^2 u=1\) with \(k\ge 2\) singular points, which was established by Gálvez–Martínez–Mira in 2005. We also obtain the existence of solutions in higher dimensional cases with general right hand sides.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    Jiguang Bao, Jingang Xiong & Ziwei Zhou

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  1. Jiguang Bao
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  2. Jingang Xiong
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  3. Ziwei Zhou
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Correspondence to Jingang Xiong.

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Communicated by O. Savin.

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All authors are supported in part by the key project NSFC 11631002, and J. Xiong is also supported in part by NSFC 11501034 and 11571019.

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Bao, J., Xiong, J. & Zhou, Z. Existence of entire solutions of Monge–Ampère equations with prescribed asymptotic behavior. Calc. Var. 58, 193 (2019). https://doi.org/10.1007/s00526-019-1639-4

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