How to prove that a functional is locally Lipscitz?

Question

Let $X$ be a real Banach space and $\Omega$ a bounded domain in $\mathbb{R}^N$. We have the following functional, for $u \in X$ and $p \in (0, \infty)$: $$F(u(x)) = \int_{\Omega}^{} \dfrac{|u(x)|^{p-1} u(x)}{|x|^p} dx, ~\forall x \in \Omega.$$ I want to prove that this functional is locally Lipschitz, satisfying the Lipschitz condition such $$|F(u)-F(v)| \leq K |u-v|~\forall u,v \in N,$$ where there is a neighbourhood $N$ of $u$ and a constant $K$ depending on $N$.\ Applying the latter, we have \begin{align*} |F(u)-F(v)| &\leq \left \vert \int_{\Omega}^{} \dfrac{|u(x)|^{p-1} u(x)}{|x|^p} dx - \int_{\Omega}^{} \dfrac{|v(x)|^{p-1} v(x)}{|x|^p} dx \right \vert \\ &\leq \left \vert \int_{\Omega}^{} \dfrac{|u(x)|^{p-1} u(x) - |v(x)|^{p-1} v(x)}{|x|^p} dx \right \vert \\ &\leq \int_{\Omega}^{} \dfrac{\left \vert |u(x)|^{p-1} u(x) - |v(x)|^{p-1} v(x) \right \vert}{|x|^p} dx. \end{align*} I got stuck in here and don't know how to proceed after this?