A direct proof of Strauss inequality in $H^1_0(\Omega)$

Question

Let $\Omega \subset \mathbb{R}^N$ a bounded domain and $u \in C^1_0(\Omega)$ such that $u(x) = u(|x|)$. Then there exists a positive constant $C$ such that $$ |u(x)| \leq C \frac{\|\nabla u\|_{L^2(\Omega)}}{|x|^{\frac{N-2}{2}}}, \quad \forall\,\, x \in \Omega. $$ I know how to prove, by density, that given $u \in H^1_0(\Omega)$, then $$ |u(x)| \leq C \frac{\|\nabla u\|_{L^2(\Omega)}}{|x|^{\frac{N-2}{2}}}, \quad \forall\,\, \text{a.e.}\,\, x \in \Omega. $$ However I am looking for a direct proof of it, I mean, without using density argument.