Variational characterisation of Neumann eigenvalue

Question

Let $1<p<\infty$ and $\Omega\subset\mathbb{R}^n$ be a bounded smooth domain. Consider the Neumann eigenvalue problem: $$ -\Delta_p u=-\text{div}(|\nabla u|^{p-2}\nabla u)=\lambda|u|^{p-2}u\text{ in }\Omega,\,\,\frac{\partial u}{\partial\nu}=0\text{ on }\partial\Omega, $$ where $\nu$ denotes the outward unit normal vector to $\partial\Omega$. Then the first nontrivial Neumann eigenvalue of the above problem can be characterised as: $$ \lambda_1=\inf_{\{u\in W^{1,p}(\Omega)\setminus\{0\}\}}\left\{\frac{\|\nabla u\|_{L^p(\Omega)}^{p}}{\|u\|^p_{L^q(\Omega)}}:\int_{\Omega}|u|^{p-2}u\,dx=0\right\}. $$ I saw in many references this characterisation is written in a straightforward way, but I could not find a proof of this fact. Can someone please help me with the same. Thanks a lot.