I want to use the contradiction argument and compact argument to prove the inequality below
$\forall\epsilon>0$,there exists $C_\epsilon>0$,$\forall u\in W^{1,p}(U)$,we have
$||u||_{L^p(U)}\leqslant\epsilon||Du||_{L^p(U)}+C_\epsilon|\bar u|$
where $\bar u:=\frac{1}{|U|}\int_U udx$.
PS:By the contradiction argument and compact argument,I can prove that there exist limit $u\in W^{1,p}(U)$ with $\bar u=0,||u||_{L^p(U)}=1$,but I failed when I want to find out a contradiction.