Existence of solutions for the Dirichlet problem in unbounded domains

Question

Suppose we are trying to solve the Dirichlet problem in a possibly unbounded domain $\Omega \subseteq \mathbb R ^n$ with continuous prescribed boundary data $f$. When $\Omega$ is bounded, it is well known that provided $\partial \Omega$ is nice enough (e.g. for each point there is a sphere touching it only at that point), a solution exists. One makes use of Perron's method to show this, and the boundedness of $f$ on $\partial \Omega$ plays a seemingly crucial role. I am interested in the case where $\Omega$ is unbounded and $f$ is not necessarily bounded on the boundary. Supposing $\partial \Omega$ is nice as before, will the Dirichlet problem generally admit a solution? Are there counterexamples?

Answer 1

As long as the complement of $\Omega$ has interior points, you can use an inversion in a sphere in the complement (using the Kelvin transform) to reduce this to the case of bounded domains with not necessarily continuous or even bounded boundary data.