## A bifurcation theorem for critical points of variational problems.(English)Zbl 0659.58007

The authors generalize bifurcation theorems for potential operators by admitting a nonlinear dependence on the bifurcation parameter. The main result is the following: Let E be a real Hilbert space (a generalization to some special Banach spaces is possible), let $$F: E\times {\mathbb{R}}\to {\mathbb{R}}$$ be $$C^ 2$$ and denote $$f(x,\lambda):=D_ xF(x,\lambda),\quad A(\lambda):=D_ xf(0,\lambda).$$ Suppose $$f(0,\lambda)\equiv 0$$ and let 0 be an eigenvalue of A(0) of a finite multiplicity and an isolated point in the spectrum $$\sigma(A(0))$$. Suppose, moreover, that for $$\lambda\neq 0$$ small enough all eigenvalues of $$A(\lambda)$$ are different from zero and the limits $$r^{\pm}:=\lim_{\lambda \to 0\pm}r(A(\lambda))$$ exist, where $$r(A(\lambda)$$) denotes the number of negative eigenvalues of $$A(\lambda)$$ converging to zero as $$\lambda\to 0$$. Then (0,0) is a bifurcation point of the equation $$f(x,\lambda)=0$$. This result is proved by the use of the center manifold theorem and the Conley index.
Reviewer: P.Quittner

### MSC:

 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces 35B32 Bifurcations in context of PDEs

### Keywords:

bifurcation; center manifold; Conley index
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### References:

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