## Isolated singularity of elliptic function

Let in an exterior domain , being uniformly elliptic. Prove that if is bounded on one side then has a limit(possibly infinite) as .

WLOG, assume , when , here

We have .

If , then has a limit which is infinity.

Otherwise, , , , such that .

Define , note that , and . By the Harnack inequality(see Thm 3.10, need a little bit of work)

, ,

By maximum principle

, ,,

So , since is arbitrary,, we know which means . So has a limit as .

Gilbarg Trudinger’s book. Chapter 3, exercise 3.3.

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