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Partial symmetry and asymptotic behavior for some elliptic variational problems. (English) Zbl 1274.35026

Summary: A short elementary proof based on polarizations yields a useful \((new)\) rearrangement inequality for symmetrically weighted Dirichlet type functionals. It is then used to answer some symmetry related open questions in the literature. The non symmetry of the Hénon equation ground states as well as their asymptotic behavior are analyzed more in depth. A special attention is also paid to the minimizers of the Caffarelli-Kohn-Nirenberg inequalities.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35J20 Variational methods for second-order elliptic equations
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