A bifurcation theorem for critical points of variational equation (Q1913931)

Let \(E\) be a real Hilbert space, \(\lambda \in \mathbb{R}\), \(F \in C^2(E \times \mathbb{R},\mathbb{R})\). It is supposed that \(A(\lambda)x + N(x,\lambda)\) is the gradient \(D_x F(x,\lambda)\) of \(F\) and \(N(x,\lambda) = o(|x|)\) as \(x \to \theta\) uniformly for bounded \(\lambda\). Let \(0\) be an isolated eigenvalue of \(A(O)\) and \(0 < n = \dim \text{ker }A(0) < + \infty\). Some existence theorems for the equation \(A(\lambda)x + N(x,\lambda) = \theta\) are proved. The notions of center manifold and Morse-Conley index are used. Relative to general bifurcation existence theorems for potentiality cases of bifurcation equations see the works by \textit{V. A. Trenogin, N. A. Sidorov} and the reviewer [Differ. Integral Equ. 3, No. 1, 145-154 (1990; Zbl 0729.47060)] and \textit{V. A. Trenogin} and \textit{N. A. Sidorov} [`Potentiality conditions of branching equation and bifurcation points of nonlinear operators', Uzb. Math. J., No. 2, 40-49 (1992)].