On the Dirichlet and the Mixed Dirichlet-Neumann Problems in Exterior Domains

We study the properties of generalized solutions in unbounded domains and the
asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover,
we study the unique solvability of the Dirichlet and the mixed Dirichlet-Neumann biharmonic
problems in the exterior of a compact set under the assumption that generalized solutions
of these problems has a bounded Dirichlet (energy) integral with weight |x|
a
. We used the
variation principle and depending on the value of the parameter a, we obtained uniqueness
(non-uniqueness) theorems of these problems or present exact formulas for the dimension of the
space of solutions.