SOLUTIONS TO OVERDETERMINED ELLIPTIC PROBLEMS IN NONTRIVIAL EXTERIOR DOMAINS

In this paper we construct nontrivial exterior domains Ω ⊂ R N , for all N ≥ 2, such that the problem −∆u + u − u p = 0, u > 0 in Ω, u = 0 on ∂Ω, ∂u ∂ν = cte on ∂Ω, 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