# Sobolev embedding theorem example

We know by Sobolev embedding theorem, that (for $$\mathbb{N}\ni n>1$$) $$W^{1,n}(B_1)\not \subset L^\infty(B_1)$$ but what is a concrete example of such a function? That is, is there a real valued function $$u$$ defined on the unit ball in $$\mathbb{R}^n$$ such that $$u\in W^{1,n}(B_1)$$ but $$u$$ is not bounded?

It's easy to find a counterexample in two-dimensional case: $$u(x) = \log(1-\log|x|).$$ Then using polar coordiantes $$\begin{cases} x = r\cos \varphi\\ y = r\sin \varphi \end{cases}$$ one can get $$\int_{B_1(0)} |\nabla u(x)|^2 dx = 2\pi \int_0^1 r \cdot \left[ \frac{d}{dr} \log(1-\log r)\right]^2 dr =$$ $$= 2\pi \int_{0}^1 \frac{dr}{(1-\log r)^2r} < \infty,$$ but obvioulsy $$u(x)$$ is not bounded.

I think one can use this example to construct the same function for any natural $$n > 1$$.