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Finding critical points whose polarization is also a critical point. (English) Zbl 1283.35023

Summary: 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 et al. [J. Anal. Math. 96, 1–18 (2005; Zbl 1206.35086)] have proved that the critical point is axially symmetric.

MSC:

35J20 Variational methods for second-order elliptic equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
49J35 Existence of solutions for minimax problems
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian

Citations:

Zbl 1206.35086
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