We consider a bounded open subset \(\Omega \) of \({\mathbb {R}}^n\) of class \(C^{1,\alpha }\) for some \(\alpha \in ]0,1[\) and we solve the Neumann problem for the Helmholtz equation both in \(\Omega \) and in the exterior of \(\Omega \) . We look for solutions in the space of \(\alpha \) -Hölder continuous functions that may not have a classical normal derivative at the boundary points of \(\Omega \) and that may have an infinite Dirichlet integral around the boundary of \(\Omega \) . Namely for solutions that do not belong to the classical variational setting.
