A nonvariational Neumann problem for the Helmholtz equation

We consider a bounded open subset $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ and in the exterior of $\mathrm{\Omega }$. We look for solutions in the space for $\alpha$-Hölder continuous functions that may not have a classical normal derivative at the boundary points of $\mathrm{\Omega }$ and that may have an infinite Dirichlet integral around the boundary of $\mathrm{\Omega }$. Namely for solutions that do not belong to the classical variational setting.