A nonvariational Neumann problem for the Helmholtz equation

Abstract 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.