The exterior Dirichlet problem for a class of fourth order elliptic equations

In some exterior domain G of the Euclidian p-space $\text{R}$^p the Dirichlet boundary value problem is considered for the equation (L + κ²)²u = f, where L is a uniformly elliptic operator and κ is a real number different from 0. It can be shown that each solution u of this equation splits into u = x_l?_lu₁ + u₂, where u₁ and u₂ satisfy Heimholte equations. Asymptotic conditions for u are formulated by imposing Sommerfeld radiation conditions on u₁ and u₂. If u₁ and u₂ are assumed to satisfy the same radiation condition, we prove a “Fredholm alternative theorem.” If u₁ and u₂ satisfy different radiation conditions, existence and uniqueness of the solution can be shown, provided the space dimension p is greater than 2.