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)


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