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On the Monge–Ampère equation with boundary blow-up: existence, uniqueness and asymptotics

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Abstract

We consider the Monge–Ampère equation det D 2 u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) is positive in Ω and non-negative on ∂Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in \({\mathbb R}^N\) with N ≥ 2. We give asymptotic estimates of the behaviour of such solutions near ∂Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ∂Ω and \(b\equiv 0\) on ∂Ω.

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Authors and Affiliations

  1. Department of Mathematics, The Australian National University, Canberra, ACT 0200, Australia

    Florica Corina Cîrstea

  2. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso M. S. Angelo, Via Cintia, 80126, Napoli, Italy

    Cristina Trombetti

Authors
  1. Florica Corina Cîrstea
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  2. Cristina Trombetti
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Correspondence to Florica Corina Cîrstea.

Additional information

Florica Corina Cîrstea’s research is supported by the Australian Research Council. F. Cîrstea was also supported by the Programma di Scambi Internazionali dell’Università degli Studi di Napoli “Federico II”. She is grateful for the hospitality and support during her research at Università degli Studi di Napoli “Federico II” in January–February 2006.

Cristina Trombetti is grateful for the hospitality and support during her research at the Department of Mathematics of the Australian National University in July–August 2005.

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Cîrstea, F.C., Trombetti, C. On the Monge–Ampère equation with boundary blow-up: existence, uniqueness and asymptotics. Calc. Var. 31, 167–186 (2008). https://doi.org/10.1007/s00526-007-0108-7

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  • DOI: https://doi.org/10.1007/s00526-007-0108-7

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