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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 $V_0$?

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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.)

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