I want to find the Frechet derivative of the functional in Sobolev space, but when I try to find the G-derivate, I don't know how to calculate.
$$\Omega\subset\mathbb{R}^n, X=W_{0}^{1,p}(\Omega,\mathbb{R}^1), 2<p<\infty, f(u)=\int_{\Omega} |\nabla u|^p\mathrm{d}x.$$
When I calculate
$$\lim\limits_{t\to 0} \frac{f(u+th)-f(u)}{t}=\int_{\Omega}p|\nabla
u|^{p-2}(\nabla u\cdot \nabla h)\mathrm{d}x$$
where $u,h\in X$. I don't know how to put the limit sign into the integral sign.