Suppose i have a weak solution $u \in H^1_0(\Omega)$ of the problem
\begin{cases} -\Delta u = |u|^{2^*-2}u, & x \in \Omega \\ u=0, & x \in \partial \Omega \end{cases}
And $\Omega$ is as regular as you want. I now need to jump the solution to $C^2$, but the problem is that I think $f(x) = |x|^{2^*-2}x $ is not Holder continuous, so I can't apply the usual theorems. How can I override this?