### Finding critical points whose polarization is also a critical point

#### Abstract

We show that near any given minimizing sequence of paths for the mountain pass lemma,

there exists a critical point whose polarization is also

a critical point. This is

motivated by the fact that if any polarization of a critical point is also a critical point and the Euler-Lagrange equation is a second-order semi-linear

elliptic problem, T. Bartsch, T. Weth and M. Willem

(J. Anal. Math., 2005) have proved that the critical point is axially symmetric.

there exists a critical point whose polarization is also

a critical point. This is

motivated by the fact that if any polarization of a critical point is also a critical point and the Euler-Lagrange equation is a second-order semi-linear

elliptic problem, T. Bartsch, T. Weth and M. Willem

(J. Anal. Math., 2005) have proved that the critical point is axially symmetric.

#### Keywords

Symmetry of solutions of semi-linear elliptic PDEs; mountain pass lemma; general minimax principle; symmetrization; polarization;non-smooth critical point theory

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