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)<c+\epsilon.

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)<Kv(x_m), \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.

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