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The Gel’fand Problem for the Biharmonic Operator

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Abstract

We study stable and finite Morse index solutions of the equation \({\Delta^2 u = {e}^{u}}\). If the equation is posed in \({\mathbb{R}^N}\), we classify radial stable solutions. We then construct nonradial stable solutions and we prove that, unlike the corresponding second order problem, no Liouville-type theorem holds, unless additional information is available on the asymptotics of solutions at infinity. Thanks to this analysis, we prove that stable solutions of the equation on a smoothly bounded domain (supplemented with Navier boundary conditions) are smooth if and only if \({N \leqq 12}\). We find an upper bound for the Hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded Morse index, provided they are controlled in a suitable Morrey norm.

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Authors and Affiliations

  1. LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039, Amiens Cedex, France

    Louis Dupaigne & Olivier Goubet

  2. School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

    Marius Ghergu

  3. LMA, UMR CNRS 5142, Université de Pau et Pays de l’Adour, Avenue de l’Université, BP 1155, 64013, Pau Cedex, France

    Guillaume Warnault

Authors
  1. Louis Dupaigne
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  2. Marius Ghergu
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  3. Olivier Goubet
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  4. Guillaume Warnault
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Correspondence to Louis Dupaigne.

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Communicated by P. Rabinowitz

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Dupaigne, L., Ghergu, M., Goubet, O. et al. The Gel’fand Problem for the Biharmonic Operator. Arch Rational Mech Anal 208, 725–752 (2013). https://doi.org/10.1007/s00205-013-0613-0

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  • DOI: https://doi.org/10.1007/s00205-013-0613-0

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