Any function that is in $W^{1,2}_0(\Omega)$ but not in $W^{1,p}_0(\Omega)$ for any $p>2$

Question

Let $\Omega$ be an open and bounded domain in $\mathbb{R}^n$. Let $W^{1,q}_0(\Omega)$ denotes the space of all functions $u\in W^{1,q}(\Omega)$ whose trace is zero on the boundary. Can anyone think of any function that is in $W^{1,2}_0(\Omega)$ but not in $W^{1,p}_0(\Omega)$ for ANY $p>2$? I am sure there is one, but I can't think of any. Thanks!

Answer 1

I hope I do not wrong any calculation. You can try with \begin{equation} u(x)=e^{\int_0^x \frac{1}{t^{1/2}log(t)}dt} -2e^{\int_0^{1/2}\frac{1}{t^{1/2}log(t)}dt}x \end{equation} with $x\in (0,1/2)$

According to me to understand how to discover the function, it is better to solve first, this easier problem:

Find $u\in W^{1,1}_0(I)$ such that $u\notin W^{1,p}(I) $ for all $P>1$, with $I$ interval of $R$.