An Overdetermined Neumann boundary value problem with a general driving force

In this paper, we prove the existence of a family of non trivial compact subdomains $\text{Ø}$ in the manifold $\mathcal{M}={\mathbb{R}}^N\times \mathbb{R}\mathrm{/}2\pi \mathbb{Z},N\ge 2$ for which the overdetermined Neumann boundary value problem \begin{align}\label{Neumann1} \left { \begin{aligned} -\D w&=μg(w) && \qquad \text{in Ω,} \frac{\partial w}{\partialη} &=0 &&\qquad $\text{on }\mathrm{\partial }\mathrm{\Omega },$ w&=c\ne 0 &&\qquad $\text{on }\mathrm{\partial }\mathrm{\Omega }\text{,}$ \end{aligned} \right. \end{align} admits solutions for some $\mu >0$ and a $C^{1,\alpha }$ function $g:\mathbb{R}\to \mathbb{R}\mathrm{.}$ The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions $g$. By this, we generalise a recent result obtained by Fall, Weth and the first named author in \cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered.