## Isolated singularity of elliptic function

$\mathbf{Problem: }$ Let $Lu=au_{xx}+2bu_{xy}+cu_{yy}=0$ in an exterior domain $r>r_0$, $L$ being uniformly elliptic. Prove that if $u$ is bounded on one side then $u$ has a limit(possibly infinite) as $r\to \infty$.

$\mathbf{Proof: }$ WLOG, assume $u(x)\geq 0$, when $|x|\geq r_0$, here $x=(x',x'')\in\mathbb{R}^2$
We have $\displaystyle \lim\limits_{\overline{|x|\to \infty}}u(x)=c\geq 0$.
If $c=+\infty$, then $u$ has a limit which is  infinity.
Otherwise, $\forall\, \epsilon>0$, $\exists\, x_m\to \infty$, $m=1,2,\cdots$, such that $u(x_m).

Define $v(x)=u(x)-c+\epsilon$,  note that $v(x_m)<2\epsilon$, $m=1,2,\cdots$ and $Lv=0$. By the Harnack inequality(see Thm 3.10, need a little bit of work)
$v(x), $\forall\, |x|=r_m$, $m=1,2,\cdots$
By maximum principle
$v(x)<2K\epsilon$, $r_m\leq|x|\leq r_{m+1}$,$m=1,2,\cdots$,
So $\overline{\lim\limits_{x\to\infty}}u(x)\leq 2K\epsilon$, since $\epsilon$ is arbitrary,, we know $\overline{\lim\limits_{x\to\infty}}v(x)\leq 0$ which means $\overline{\lim\limits_{x\to \infty}}u(x)\leq c$. So $u$ has a limit as $r\to\infty$.
$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ Gilbarg Trudinger’s book. Chapter 3, exercise 3.3.