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Uniqueness and symmetry of minimizers of Hartree type equations with external Coulomb potential. (English) Zbl 1255.35023

Summary: In a recent paper, V. Georgiev and G. Venkov [J. Differ. Equations 251, No. 2, 420–438 (2011; Zbl 1219.35086)] establish first radial symmetry and then uniqueness of minimizers to a certain functional. In the present paper we prove first the uniqueness of possible positive minimizers by revealing a hidden convexity property of the underlying functional. Then symmetry follows from the simple observation that uniqueness fails if there is a nonradial minimizer, because it could be rotated and give rise to a second minimizer.

MSC:

35B06 Symmetries, invariants, etc. in context of PDEs
35B09 Positive solutions to PDEs
35J20 Variational methods for second-order elliptic equations
35Q40 PDEs in connection with quantum mechanics

Citations:

Zbl 1219.35086
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