This problem basically arises when I read Gilbarg and trudinger’s PDE books theorem 8.26. To avoid too many notations, I rewrite the problem as follows:
Given an open bounded and connected subset $\Omega\subset R^n$, let $u\in W^{1,2}(\Omega)$, now if the following holds: $$ M:=\sup_{\partial \Omega\cap B_{2R}}u^+, \text{ here } B_{2R}\text{ is any ball in }R^n, \ u_M^+= \sup{u,M}\text{ if }x\in \Omega; u_M^+=M, \text{ if }x \notin \Omega. $$ And $$ \bar{u}=u_M^++k,\ v=\eta^\beta(\bar{u}^\beta-(M+k)^\beta),\beta>0,k \text{ is some positive constant, } \eta \in C^1_0(B_{4R}) $$ We further have $$ \sup_{\partial \Omega} u=\inf _{\partial \Omega}\{k|(u-k)^+\in W_0^{1,2}(\Omega)\} $$
Now my question is : if $\bar{u}$ is bounded, how can we show $v\in W_0^{1,2}(B_{2R})$?
(it is natural for $ v \in W_0^{1,2}(\Omega\cap B_{2R})$.)
A picture shows the relationship between $\Omega$ and the ball.
