Symmetry of solutions to semilinear elliptic equations via Morse index

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On the symmetry of minimizers (2007, methodology): “We have to mention that in a series of recent papers (see [2], [15], [17] and references therein), different new techniques were developed to study the symmetry of solutions for some classes of elliptic problems.”
Symmetry of Solutions to Variational Problems for Nonlinear Elliptic Equations via Reflection Methods (2010, methodology): “We merely remark that in this case we are not able to find e ∈ S such that λ1(e, Ves) ≥ 0. Instead, we use the even potential Ves in another way to find e ∈ S such that λ1(e, Ve) is nonnegative, see [52, 72 ].”