Let $u(x)=\alpha+\beta U(x)$, where $U(x)=|x|^{2-m}$ ($N\geq3$) is the fondamental harmonic function, $\alpha<0$ and $\beta>0$. We know that $u$ is superharmonic on $\mathbb{R}^m$ and harmonic on $\mathbb{R}^m\setminus\{0\}$. Let $a>0$ be a given constant. Is it possible to modify $u$ on a neighborhood of infinity to obtain a function $\overline{u}$ that is $=u$ for $|x|<R$, superharmonic on $\mathbb{R}^m\setminus\{0\}$ and such that $\lim \overline{u}(x)\geq a$, as $x\to\infty$, for some $R>0$?

I tried to find by "bricolage" of adding some constant here and subtracting some other there, but in vain.