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\(\mathbb{Z}_2\)-equivariant Ljusternik-Schnirelman theory for non-even functionals. (English) Zbl 0907.58006
Starting off with the classical result which asserts that any smooth and even functional defined on the sphere \(S^n\) has at least \(n+1\) pairs of antipodal critical points, the authors develop an equivariant Ljusternik-Schnirelman theory for non-even functionals. The main result of this paper shows that if a \({\mathbb{Z}}_2\)-equivariant min-max procedure is applied to a non-symmetric functional \(\varphi\), then one obtains either the usual critical points, or a new class of points \(x\), defined by \(\varphi (x)=\varphi (-x)\) and \(\varphi '(x)=\lambda \varphi '(-x)\), for some \(\lambda >0\). The proof is based on the observation that the equivariant Ljusternik-Schnirelman min-max levels for the original functional \(\varphi\) and for the even functional \(\psi (x)=\max \{\varphi (x),\varphi (-x)\}\) are the same.
The results are deep, very interesting and it seems that they may be applied in a wide class of situations. The authors illustrate their abstract results in order to prove a bifurcation-type result for a class of non-homogeneous semilinear elliptic boundary problems with subcritical nonlinearity.

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E09 Group-invariant bifurcation theory in infinite-dimensional spaces
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