Squassina, Marco; Van Schaftingen, Jean Finding critical points whose polarization is also a critical point. (English) Zbl 1283.35023 Topol. Methods Nonlinear Anal. 40, No. 2, 371-379 (2012). 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. Cited in 2 Documents 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 Keywords:symmetry of solutions of semi-linear elliptic PDEs; mountain pass lemma; general minimax principal; symmetrization; polarization; non-smooth critical point theory Citations:Zbl 1206.35086 PDFBibTeX XMLCite \textit{M. Squassina} and \textit{J. Van Schaftingen}, Topol. Methods Nonlinear Anal. 40, No. 2, 371--379 (2012; Zbl 1283.35023) Full Text: arXiv