Suppose $U$ is an open set in $\mathbb{R}^{n}$ ($n\geq2$) whose complementary is not polar, and $f$ is a real-valued function defined at least on the boundary of $U$. We know that the generalized Dirichlet problem has a solution, i.e., if $f$ is continuous and all boundary points of $U$ are regular, there is a function $H\_{f}^{U}(x)$ that is harmonic on $U$ and tends to $f(y)$, as $x\to y$, for all $y$ in the boundary of $U$. The same is also true if $f$ is superharmonic, but not necessarily continuous, on a neighborhood of the boundary of $U$. 


Suppose now that $U$ is an open set in $\mathbb{R}^{n}\cup \{\infty\}$, the one point compacification of $\mathbb{R}^{n}$. Do the same results hold for $f$ continue , or $f$ superharmonic (not necessarily continuous)? Do you know reference for that stuff?



A question on the problem of Dirichlet
I found a book that talks about the problem of Dirichlet for unbounded regions. This is Lester L. Helms' book on " potential theory", Springer, 2009, Chapter 5. According to this book, the answer to the first question is yes; i.e. if $f$ is continuous on the boundary of $U$ then $H^{U}\_{f}(x)$ is harmonic and tends to $f(y)$, as $x\to y$, for all regular point $y$ in the boundary of $U$. But steel I do not know the answer to the second question.


