Let $\Omega\subseteq\mathbb{R}^n, 2<p<\frac{2n}{n-2}$, $E: H^1_0(\Omega)\to\mathbb{R}$, $$E(u) = \frac{1}{p}\int_\Omega |u|^p dx$$.
I want to show that the Fréchet-derivative of $E$ in a point $u$ is given by $$DE(u)v = \int_\Omega uv|u|^{p-2} dx$$
I have to show that $$\lim\limits_{||v||\to 0} \frac{1}{||v||} |\frac{1}{p} \int_\Omega |u+v|^p - |u|^p - puv|u|^{p-2}dx = 0.$$
I tried around a bit: $$\begin{align} |u+v|^p-|u|^p-puv|u|^{p-2} &= (u+v)(u+v)|u+v|^{p-2}-|u|^p-puv|u|^{p-2} \\&= (|u|^2 +2uv + |v|^2)(|u+v|^{p-2} - |u|^p -puv|u|^{p-2} \\&\leq (|u|^2 +2uv + |v|^2)(|u|^{p-2}+|v|^{p-2} - |u|^p -puv|u|^{p-2} \\&= |u|^2|v|^{p-2} + 2uv(|u|^{p-2} + |v|^{p-2}) + |v|^2|u|^{p-2} + |v|^p -puv|u|^{p-2} \\&= (2-p)uv|u|^{p-2} + (|u|^2+|v|^2 + 2uv)|v|^{p-2} + |v|^2|u|^{p-2} \end{align}$$
But so far this doesn't help. Could someone give me a hint on that?
