Variational properties of a nonlinear elliptic equation and rigidity

Details Variational properties of a nonlinear elliptic equation and rigidity Variational properties of a nonlinear elliptic equation and rigidity

1999-02-02

arxiv.org/abs/math/9902016v1

Mathematics

Dynamical Systems

11 pages, no figures

Scientific paper

We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally minimising. However, in dimension two the situation is different. We show that for radially symmetric potentials there always exist solutions which are not locally minimal unless the potential vanishes identically. We discuss the relations of this with the so-called Hopf rigidity phenomenon.

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