I have to show that the function $u(x)=(-log|x|)^{\frac{N-1}{N}}$ is not in the Sobolev space $W^{1,N}$.
My guess:
I have tried to show the "Trudinger's inequality" $\left( u(x)\in W^{1,N} \Rightarrow \int e^{|u|^{\frac{N-1}{N}}}<\infty \right)$ does not hold, therefore $u(x)$ won't be in $W^{1,N}$. But, when you manipulate a little bit, you arrive to the conclusion that the integral is equal to $1$, so you cannot conclude that $u(x)$ is not in $W^{1,N}$.
How would you show that? Maybe obtaining the gradient ($\nabla$) of $u(x)$ and computing the integral of the modulus of $\nabla u$ to the $N$-th power to see if it is in the required space, i.e: $\int\lvert\nabla u\lvert^N<\infty$? I think it is a long path and (if I remember) you do not arrive anywhere.
Any guesses, ideas or hints?