A degenerate bifurcation from simple eigenvalue theorem (Q2127476)

This paper is concerned with the abstract bifurcation problem on Banach spaces with a real parameter \(\lambda\): \(F(\lambda, u)=0\). Consider the local bifurcation near \((\lambda_0,u_0)\) and assume that \(0\) is a simple eigenvalue of the linearized operator \(F_u(\lambda_0, u_0)\). Recall two crucial conditions and their complements: \((\mathbf{F3})\) \(F_{\lambda u}(\lambda_0, u_0)[w_0] \notin R(F_u(\lambda_0, u_0))\); \((\mathbf{F4})\) \(F_{u u}(\lambda_0, u_0)[w_0]^2 \notin R(F_u(\lambda_0, u_0))\); \((\mathbf{F3}')\) \(F_{\lambda u}(\lambda_0, u_0)[w_0] \in R(F_u(\lambda_0, u_0))\); \((\mathbf{F4}')\) \(F_{u u}(\lambda_0, u_0)[w_0]^2 \in R(F_u(\lambda_0, u_0))\). Under suitable assumptions, the problem is classified into four cases according to different combinations of the above conditions, which lead to different types of bifurcation phenomema: (a) Transcritical: \((\mathbf{F3})\) and \((\mathbf{F4})\), a crossing curve of nontrivial solutions; (b) Pitchfork: \((\mathbf{F3})\) and \((\mathbf{F4}')\), a crossing curve of nontrivial solutions bending leftward or rightward; (c) Tangential: \((\mathbf{F3}')\) and \((\mathbf{F4})\), a tangential curve of nontrivial solutions bending upward or downward; (d) Double transcritical: \((\mathbf{F3}')\) and \((\mathbf{F4}')\), two crossing curves of nontrivial solutions. Cases (a) and (b) are well-known in [\textit{M. G. Crandall} and \textit{P. H. Rabinowitz}, J. Funct. Anal. 8, 321--340 (1971; Zbl 0219.46015)], Case (d) has been proven in [\textit{P. Liu} et al., J. Funct. Anal. 264, No. 10, 2269-2299 (2013; Zbl 1282.35043)], and Case (c) is just the main concern of the present paper. Apart from a degenerate bifurcation theorem, the stability of bifurcating solutions is also obtained and two examples of semilinear elliptic PDEs are given to illustrate applications of the theorems.