Solutions to overdetermined elliptic problems in nontrivial exterior domains

Source abstract

In this paper we construct nontrivial exterior domains \Omega \subset \mathbb R^N , for all N\geq 2 , such that the problem \left\{\begin{array} {ll} -\Delta u +u -u^p=0,\ u >0 & \text{in }\; \Omega, \\ \ u= 0 & \text{on }\; \partial \Omega, \\ \ \frac{\partial u}{\partial \nu} = \text{cte} & \text{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.