Let $$ X=\{u\in W^{1,p}(\Omega):\int_{\Omega}|u|^{q-2}u\,dx=0\}, $$ where $1<p<N$ and $1<q<p^{*}=\frac{Np}{N-p}$. Here $\Omega$ is a bounded domain.
Here $X$ is endowded with the norm $$ \|u\|=\|u\|_{L^p(\Omega)}+\|\nabla u\|_{L^p(\Omega)}. $$ My question is it possible to say that $$ \|u\|_1=\|\nabla u\|_{L^p(\Omega)} $$ is an equivalent norm on $X$? I hope if this is true, the some version of Poincare inequality might help which I am unaware. Kindly help. Thanks in advance.