Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity

In this paper, we first apply the implicit function theorem to find nontrivial solutions to the following Bernoulli’s overdetermined problem \begin{equation} -Δu=γ,, $\text{in}$,,Ω\setminus\overline{D}, ,, u=1,,,\partial_$\mathbf{n}$ u=Q,,$\text{on}$,,\partial D, ,,u=0,,$\text{on}$,,\partialΩ\nonumber \end{equation} for a fixed boundary $\mathrm{\partial }\mathrm{\Omega }$ and a free boundary $\mathrm{\partial }D$ in ${\mathbb{R}}^N$ with $N\ge 2$, where $\gamma$ is a sufficiently small constant, the Bernoulli value $Q$ is determined by the construction, and $\mathrm{\Omega }\setminus \bar{D}$ is a bounded ring-shaped domain. In dimension $N=n+1$ with $n\ge 2$, we further study the existence of a nontrivial unbounded domain $\mathrm{\Omega }\setminus \bar{D}$ in ${\mathbb{R}}^N$ such that the Bernoulli’s overdetermined problem admits a solution. This unbounded domain we construct is periodic in one variable and radial in the other variables by bifurcation analysis.