# Dirichlet problem for a subharmonic function

Suppose $$K$$ is a compact subset of $$\mathbb R^n$$ , $$V_0$$ and $$V_1$$ the complements of $$K$$ in $$\mathbb R^n$$ a and $$\mathbb R^n_\infty$$ (one point compactification), respectively. Let $$u$$ be subharmonic on $$V_0$$ and $$H$$ be the generalized solution of Dirichlet problem of $$u$$ on $$V_1$$. So in particular $$H$$ is harmonic on $$V_1$$; meaning this is harmonic in the usual sense on any open subset of $$V_1$$ that does not contain infinity, and if $$W$$ is an open subset of $$V_1$$ that contains infinity, then $$H$$ is continuous at infinity and $$H(\infty)$$ equals the mean value of $$H$$ over any ball $$B$$ whose closure is contained in $$W$$ ( see Helms, « introduction to potential theory », chapter on Dirichlet problem for unbounded domains). My question is: can we say $$u\leq H$$ on $$V_0$$?

The examples given in Helms's book already answer your question in the negative: if $$K$$ is the unit ball and we prescribe zero boundary values, then we have $$H(x) = 0,$$ but we can have $$u(x) = c (1 - |x|^{2 - n})$$ for any $$c \in \mathbb{R}$$. (If $$n = 2$$, set $$u(x) = c \log |x|$$ instead.)