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