Inequality in Sobolev spaces: does it hold?

Question

Let $\Omega$ be an open bounded subset of $\mathbb{R^n}$ and let $1\leq p < n$. Consider $$ X = W_{0}^{1, p}(\Omega)\cap L^{\infty}(\Omega).$$ Let $u\in X$ and $l> p^{\ast}$. Does the inequality $$\Vert u\Vert_{\infty}^{l-p} \Vert u\Vert_{W_0^{1,p}}^p \leq\Vert u\Vert_{X}^l$$ hold? Could anyone explain me why? I tried by using Holder inequality but I didn’t solve the problem. Could anyone please help? Thank you in advance!