We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space \(\mathbb {R}^N\) , \(N\ge 4\) . These domains arise as perturbations of complements of straight cylinders in \(\mathbb {R}^N\) , and by definition they support a positive harmonic function with vanishing Dirichlet boundary values and constant Neumann boundary values, the so-called roof function. While the domains have a similar shape as those constructed in the recent work [18] for the case \(N=3\) , there is a striking contrast with regard to the shape of corresponding roof functions which are bounded for \(N \ge 4\) . Moreover, while the analysis in [18] does not extend to higher dimensions, the approach of the present paper depends heavily on the assumption \(N \ge 4\) .
