I'm studying Sobolev spaces and I have to demonstrate that
$ u(x)=(log\frac{1}{|x|})^\alpha \in W^{1,N}(B_\frac{1}{2}) $ and $ u(x) \not\in L ^\infty(B_\frac{1}{2}), $ for $ 0<\alpha<1-\frac{1}{N}.$
(This is a counter example of the fact that $ W^{1,N}(\mathbb{R}^N) \hookrightarrow L^\infty (\mathbb{R}^N)$ is not true.)
I proved the fact $ u(x) \not\in L ^\infty(B_\frac{1}{2}) $, but I don't know how to demonstrate $ u(x) \in W^{1,N}(B_\frac{1}{2}) $
First I have to prove that $ u \in L^N . $ I have to calculate
$ \int_{B_\frac{1}{2}} |(log\frac{1}{|x|})|^{\alpha*N} \,dx = \int_{0}^{\frac{1}{2}} \int_{∂B_\frac{1}{2}} |(log\frac{1}{|\rho|})|^{\alpha*N} \,d\sigma \,d\rho $ = ?