Fréchet derivative of non-coercive energy functional $\frac{1}{2}\int_\Omega |\nabla u|^2 - \frac{1}{p}\int_\Omega |u|^p$ Let $\Omega \subseteq \subseteq \mathbb{R}^n$, $n \geq 3$, and for $2 \leq p \leq 2^* := 2n/(n - 2)$ define $E \colon H^1_0(\Omega) \to \mathbb{R}$ by 
$$E(u) := \frac{1}{2}\int_\Omega |\nabla u|^2 - \frac{1}{p}\int_\Omega |u|^p.$$
I want to show that $E$ is Fréchet differentiable. First of all, we compute the Gâteaux derivative as follows.
$$\frac{d}{d\varepsilon}\bigg\vert_{\varepsilon = 0} E(u + \varepsilon v) = \int_\Omega \nabla u\nabla v - \int_\Omega u|u|^{p - 2}v.$$ Thus a good choice for the Fréchet derivative $dE(u) \in (H^1_0(\Omega))^*$ would be
$$dE(u)(v) := \int_\Omega \nabla u\nabla v - \int_\Omega u|u|^{p - 2}v.$$ 
Then we compute
$$E(u + v) - E(v) - dE(u)(v) = \frac{1}{2}\int_\Omega|\nabla v|^2 - \frac{1}{p}\int_\Omega\left(|u + v|^p - |u|^p\right) + \int_\Omega u|u|^{p - 2}v.$$
If we let $\|v\|_{H^1_0(\Omega)} \to 0$, the first term is no problem, however, I do not know how to handle the second and the third term. A friend of mine suggested to use Taylor, but I do not see how.
Thank you!

It should first be noted that $E$ is well-defined thanks to the Sobolev embedding $$W^{k,p}(U)\subset L^{\frac{np}{n-kp}}(U)$$ for bounded, open $U\subset\mathbb R^n$ and $1\le k < \frac np$.
Now, using Hölder's inequality and the Sobolev inequality $$\lVert u\rVert_{L^{\frac{np}{n-kp}}(U)}\lesssim \lVert u\rVert_{W^{k,p}(U)},$$ valid for $1\le k < \frac np$, with $k=1, p=2$, we get
$$\left\lvert\int_\Omega u|u|^{p - 2}v\right\lvert\le\left(\int_{\Omega}\left(\lvert u\rvert^{p-1}\right)^{\frac{p}{p-1}}\right)^\frac{p-1}p \lVert v\rVert_{L^p} = \lVert u\rVert_{L^p}^{p-1}\lVert v\rVert_{L^p}\lesssim \lVert u\rVert_{L^p}^{p-1}\lVert v\rVert_{H^1}\to 0.$$
For the second term, use
$$\lvert a+b\rvert^p-\lvert a\rvert^p\le np \lvert a\rvert^{p-1} \lvert b\rvert$$ for $a,b\in\mathbb R^n$ so that by the same argument as before $$\frac 1p\int_\Omega |u + v|^p - |u|^p\le n \int_\Omega \lvert u\rvert^{p-1}\lvert v\rvert\lesssim\lVert u\rVert_{L^p}^{p-1}\lVert v\rVert_{H^1}\to 0.$$
