Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by:
$$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where $p <\frac{2n}{n-2}$. Let $v_0$ be the unique maximizer of $J$ over the set $\mathcal{A} = \{v \in \mathscr{H}_0^1 : \|v\|_{\mathscr{H}_1} \leq 1\}$ (the existence of this could be shown using, say, Rellich Kondrachov). Now, I would like to show which PDE this maximizer solves. The usual method is to compute the first variation and then from there obtain the Euler Lagrange Equation for the variational problem. However, this doesn't seem to work, as $J$ has no derivatives of $v$ in it. Specifically, if we fix some test function $\phi \in C_c^\infty(\Omega)$, and perturb by $t$ and compute the derivative, we get:
$$ \frac{d}{dt}\int_\Omega |(v + \phi t)|^p = \int_\Omega \frac{d}{dt}|v + (\phi t)|^p = \int_\Omega p|v + (\phi t)|^{p-1}\phi $$ (side note, why are we allowed to pass the derivative under the integral in the general case of the Lagrangian?). Evaluating the variation at time $t = 0$ and setting equal to $0$, we obtain: $$ \int_\Omega p|v|^{p-1}\phi = 0 $$ But this doesnt give us any useful information. What am I doing wrong here? How should I set up the variational problem?