Nonnegative superharmonic function is constant for dimension d=2?

Question

Suppose that $f\in C^2(\mathbb{R}^2)$ satisfies $\Delta f\le0$ and $f\ge0$, prove that $f$ is constant.
It seems that it's not a difficult question, but how should I consider the usage of the functions $g_\varepsilon(x)=f(x)\pm\varepsilon\ln|x|$ ($\varepsilon>0$) and combining the minimum principle of $f$ or $g_\varepsilon$ (or consider the other combinations of $f$ and the fundamental solution of $\Delta$) ?
Remark: for dimension $d\ge3$, $f$ can be a nonconstant function, for example, let $f=\frac{1}{A+|x|^{d-2}}$ where $A>0$.
for dimension $d=1$, we can prove this proposition by the properties of a convex function.

Answer 1

This question has been resolved,see (https://mathoverflow.net/questions/101564/question-about-harmonic-function-theory)