I am looking for a simple (intuitive) example of a function $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is an open set and, obviously, $p \leq N$.
Sobolev embedding theorem asserts that this can not be found for $N < p$, but I want to understand intuitivelly what this means.
For $p<n$, the function $$f(x)=\|x\|^{-\epsilon}$$ works (on the unit open ball, for example), when $\epsilon>0$ is sufficiently small. How small? Well, the gradient has one degree of homogeneity less, so $|\nabla f(x)|=O(\|x\|^{-\epsilon-1})$. And we want this to be in $L^p$, which requires $p(-\epsilon-1)>-n$. Rearrange to get $$\epsilon<\frac{n}{p}-1$$
The power functions don't work for $p=n$; one needs a milder blow-up then. Even the logarithm $\log\frac{1}{\|x\|}$ fails to be in $W^{1,n}$ since its gradient is $\sim \|x\|^{-1}$, not in $L^n$. But slower blow-up such as $$\log\log \frac{1}{\|x\|}$$ or even $$\left(\log \frac{1}{\|x\|}\right)^p,\quad 0<p<1$$ allows the function to be in $W^{1,n}$.
At some point, one should also affirm that the pointwise gradient that I've been referring to agrees with the distributional gradient of these functions. But since you're looking for simple and intuitive things, I omit these details.