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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:
 5.8e+06 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 5.8e+10 Group-invariant bifurcation theory in infinite-dimensional spaces
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References:
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