Sobolev embedding theorem example

Question

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?

Answer 1

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 $n > 1$.