Symmetrization and minimax principles

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Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent (2017, methodology): “In the case N ≥ 2, under the additional assumptions that the function f is odd and has constant sign on (0, +∞), we obtain a radially symmetric solution to (C∗) by the Schwarz symmetrization [32] and the symmetric variational principle [32, Theorem 3.”
Existence of groundstates for a class of nonlinear Choquard equations (2012, background): “With this additional assumption, Therefore, by the symmetric variational principle [40, theorem 3.2], we can prove as in the proof of Proposition 2.1 the existence of a sequence ( u n ) n ∈ N and ( v n ) n ∈ N such that as n → ∞ , I ( u n ) → b, where u ∗ n : R N → R is the Schwarz symmetrization…”