I am reading the following paper: Takáč, Peter On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51 (2002), no. 1, 187–237.
I need help to understand the following argument (page 193 section 2.1):
$\Omega\subset\mathbb{R}^N$ is a bounded regular domain, $p\in (2,\infty)$. Let $\phi_1$ be the first eigenfunction associated with the problem $-\Delta_p u=f$ and $u\in W_0^{1,p}(\Omega)$, i.e. $$\int|\nabla\phi_1|^{p-2}\nabla\phi_1\nabla v=\lambda_1\int|\phi_1|^{p-2}\phi_1v,\ \forall\ v\in W_0^{1,p}$$
where $\lambda_1>0$ os the first eigenvalue. We can assume that $\phi_1\in C^1(\overline{\Omega})$, $\phi_1>0$ in $\Omega$ and $\frac{\partial\phi_1}{\partial\eta}<0$ in $\partial\Omega$, where $\frac{\partial\phi_1}{\partial\eta}$ represents derivative in the normal direction.
Define in $W_0^{1,p}$ the semi-norm $$\|u\|_{\phi_1}=\Big(\int|\nabla\phi_1|^{p-2}|\nabla u|^2\Big)^{\frac{1}{2}}$$
The author says that in fact, $\|\cdot\|_{\phi_1}$ is an norm because of the following argument: if $v\in W_0^{1,p}$, then
\begin{eqnarray} \lambda_1\int\phi_1^{p-1}v^2 &=& \int|\nabla\phi_1|^{p-2}\nabla\phi_1\nabla(v^2) \nonumber \\ &\leq& 2\int|\nabla\phi_1|^{p-1}|\nabla v||v| \nonumber \\ &\le& 2\|v\|_{\phi_1}\Big(\int|\nabla\phi_1|^pv^2\Big)^{\frac{1}{2}} \end{eqnarray}
I can understand the last two inequalities, and I can use it to prove that $\|\cdot\|_{\phi_1}$ is a norm. The problem is the first equality. To use the characterization of the eigenvalue, we have to take $u\in W_0^{1,p}$. Why does $v^2\in W_0^{1,p}$? I think that he is using this fact, is this true?