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A Paneitz–Branson type equation with Neumann boundary conditions

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Published/Copyright: November 8, 2019

Abstract

We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.


Communicated by Frank Duzaar


Award Identifier / Grant number: 339958

Funding statement: The authors were supported by PDR T.1110.14F (FNRS). D. Bonheure was partially supported by the project ERC Advanced Grant 2013 no. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT”.

Acknowledgements

The authors thanks Bruno Premoselli for many helpful discussions on the topics of the current paper.

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Received: 2019-03-25
Accepted: 2019-10-08
Published Online: 2019-11-08
Published in Print: 2021-10-01

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