## 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

### Citations:

Zbl 0219.46015; Zbl 0275.47044; Zbl 0949.47050