JournalsjemsVol. 22, No. 1pp. 253–281

Solutions to overdetermined elliptic problems in nontrivial exterior domains

  • Antonio Ros

    Universidad de Granada, Spain
  • David Ruiz

    Universidad de Granada, Spain
  • Pieralberto Sicbaldi

    Universidad de Granada, Spain and Aix Marseille Université, Marseille, France
Solutions to overdetermined elliptic problems in nontrivial exterior domains cover

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Abstract

In this paper we construct nontrivial exterior domains ΩRN\Omega \subset \mathbb R^N, for all N2N\geq 2, such that the problem

{Δu+uup=0, u>0\mboxin  Ω, u=0\mboxon  Ω, uν=\mboxcte\mboxon  Ω,\left\{\begin{array} {ll} -\Delta u +u -u^p=0,\ u >0 & \mbox{in }\; \Omega, \\[1mm] \ u= 0 & \mbox{on }\; \partial \Omega, \\[1mm] \ \frac{\partial u}{\partial \nu} = \mbox{cte} & \mbox{on }\; \partial \Omega, \end{array}\right.

admits a positive bounded solution. This result gives a negative answer to the Berestycki–Caffarelli–Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found in the literature; however, our example is the first one in the form of an exterior domain.

Cite this article

Antonio Ros, David Ruiz, Pieralberto Sicbaldi, Solutions to overdetermined elliptic problems in nontrivial exterior domains. J. Eur. Math. Soc. 22 (2020), no. 1, pp. 253–281

DOI 10.4171/JEMS/921