Let is a bounded domain. Fix , is a function in such that , where is the distance function. Then

has a unique solution satisfying

where the constant depends only on and .

Firstly, recall some basic properties of distance function. By lemma 14.16 in GT’s book, if is and bounded, there is a neighborhood of in , say , such that . And is bounded in .

is means there exists such that is in for any .

Suppose with . Denote .

Fix ,

if small enough.

Since is compact, there exists finitely many such that covers . Let is solution of in and on .

Define , then on and in . So

in and on

Consequently, by the maximum principle,

Since has an upper bound only depends on the geometry of and , we only need to prove when is near the boundary, where . Note that

uniformly as

So there exists a neighborhood of such that and

in and on

By the maximum principle, in .

Gilbarg, Trudinger. Chapter 4, exercise 4.6. p71

Also see, J.H. Michael. A general theory for linear elliptic partial differential equations. 1977

Gary M. Lieberman. Elliptic equations with strongly singular lower order terms. 2008