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


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