I have a problem with the following inequality. Suppose r and r' are Holder conjugate, i.e. $\frac{1}{r} + \frac{1}{r'} = 1$. Then we have
$$|||u|^{\alpha -1} u||_{W^{1, r'}(\mathbb{R}^n )} \leq c ||u||_{L^r(\mathbb{R}^n)}^{\alpha -1} ||u||_{W^{1, r}(\mathbb{R}^n)}.$$
I tried with the standard Holder inequality but I failed. Can someone help me, please?