An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces

Abstract:

The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations $Ax + \varphi _\lambda (x) = \lambda x$ with $\varphi$ continuous jointly in $(\lambda ,x)$ and $\varphi _\lambda (\cdot )$ of class $C^{1,1}$. We also prove a bifurcation theorem for critical points of the function $g_{\lambda }(x)$ which is just continuous and changes at $x=0$ an isolated minimum (in $x$) to isolated maximum when $\lambda$ passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.