Let $\Omega$ be a compact subset in $\mathbb{R}^n$, and $p<n,\frac{1}{p*}=\frac{1}{p}-\frac{1}{n} $. The usual Sobolev ineuqality says that \begin{equation} \|u\|_{p*}\leq C (\|u\|_p+\|\nabla u\|_p) \end{equation} for $u\in W^{1,p}(\Omega)$. My question is, does the following stronger inequality holds: \begin{equation} \|u\|_{p*}\leq C (\|u\|^{\theta}_p\|\nabla u\|^{1-\theta}_p) \end{equation} where $0<\theta<1$.
Thanks for anyone who could help.