×

The generalized saddle-node bifurcation of degenerate solution. (English) Zbl 1124.47045

Consider the bifurcation problem for the equation \(F(u, \lambda)=0,\) where \(\lambda\) is a parameter, \(F: X\times \mathbb R\to Y\) is a \(C^1\) mapping, \(X,Y\) are Banach spaces. We recall that \((u_0, \lambda_0)\) is called a degenerate solution of \(F(u,\lambda)=0\) if \(F(u_0, \lambda_0)=0\) and \(N(F_u(u_0, \lambda_0))\neq \{0\}\). In the case \(\dim N(F_u(u_0, \lambda_0))= \) codim \(R(F_u(u_0, \lambda_0))=1,\) the bifurcation of the degenerate solution is studied in the works of M. G. Crandall and P. H. Rabinowitz [J. Funct. Anal. 8, 321–340 (1971; Zbl 0219.46015); Arch. Ration. Mech. Anal. 52, 161–180 (1973; Zbl 0275.47044)] and of J. P. Shi [J. Funct. Anal. 168, 494–531 (1990; Zbl 0949.47050)].
In this paper, the authors extend the results of Crandall and Rabinowitz for the case \[ \dim N(F_u(u_0, \lambda_0))\geq \text{codim}\;R(F_u(u_0, \lambda_0))=1. \] The tools to prove the results are the generalized inverse and the implicit function theorem.
Reviewer: Anh Cung (Hanoi)

MSC:

47J15 Abstract bifurcation theory involving nonlinear operators
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
PDF BibTeX XML Cite