Let us define the sets: $$ A=\{u\in W^{1,p}(\Omega):\int_{\Omega}u\,dx=0\} $$ and $$ B=\{u\in W^{1,p}(\Omega):\int_{\Omega}|u|^{q-2}u\,dx=0\}. $$ Here $W^{1,p}(\Omega)$ denotes the usual Sobolev space, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$, $1<p<n$ and $1<q<p^*=\frac{np}{n-p}$. Let us define $$ \lambda_A=\inf_{u\in A}\{\int_{\Omega}|\nabla u|^p\,dx:\int_{\Omega}|u|^q\,dx=1\} $$ and $$ \lambda_B=\inf_{u\in B}\{\int_{\Omega}|\nabla u|^p\,dx:\int_{\Omega}|u|^q\,dx=1\}. $$
My question: Is $\lambda_A=\lambda_B$? I tried to understand, but I could not able to see at least one inequality. Can someone please help. Thanks.