The uniqueness and existence of solutions for the 3-D Helmholtz equation in a cylindrically symmetric waveguide with unbounded perturbation

In this paper we study the 3-D Helmholtz equation in a cylindrically symmetric waveguide with unbounded perturbation, allowing the presence of guided waves. Our assumptions on the perturbed and source terms are also few. On the basis of Green’s function for the 3-D homogeneous Helmholtz equation in a cylindrically symmetric waveguide without perturbation, we introduce a generalized (out-going) Sommerfeld–Rellich radiation condition, and then we prove the uniqueness and existence of solutions for the studied 3-D Helmholtz equation satisfying our radiation condition.