Bifurcation of sign-changing solutions for an overdetermined boundary problem in bounded domains

We obtain a continuous family of nontrivial domains $\Omega_s\subset \mathbb{R}^N$ ($N=2,3$ or $4$), bifurcating from a small ball, such that the problem \begin{equation} -\Delta u=u-\left(u^+\right)^3\,\, \text{in}\,\,\Omega_s, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega_s \nonumber \end{equation} has a sign-changing bounded solution. Compared with the recent result obtained by Ruiz, here we obtain a family domains $\Omega_s$ by using Crandall-Rabinowitz bifurcation theorem instead of a sequence of domains.