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Pré-Publication, Document De Travail Année : 2017

SOLUTIONS TO OVERDETERMINED ELLIPTIC PROBLEMS IN NONTRIVIAL EXTERIOR DOMAINS

Résumé

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.
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Dates et versions

hal-01481863 , version 1 (03-03-2017)

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  • HAL Id : hal-01481863 , version 1

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Antonio Ros, David Ruiz, Pieralberto Sicbaldi. SOLUTIONS TO OVERDETERMINED ELLIPTIC PROBLEMS IN NONTRIVIAL EXTERIOR DOMAINS. 2017. ⟨hal-01481863⟩
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