Interval bifurcation theorems for Fredholm operator and its application to an elliptic overdetermined problem in bounded domains

Abstract

We establish local interval bifurcation theorem and global interval bifurcation theorem for Fredholm operator with index $0$ via $0$-group. As one of applications, we investigate the existence of a family of nontrivial domains $\Omega_{\rho}\subset \mathbb{R}^N$ ($N=2,3$ or $4$), bifurcating from a small ball, such that the problem \begin{equation} -\Delta u=u-\left(u^+\right)^3\,\, \text{in}\,\,\Omega_{\rho}, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega_{\rho} \nonumber \end{equation} has a sign-changing bounded solution. Compared with the recent result \cite[Theorem 2.1]{Ruiz}, here we obtain a family of domains $\Omega_{\rho}$ instead of a sequence of domains.