Let $I=[0,1]$ and consider the function $F:H^1(I)\to \mathbb R$ given by $$F(u)=\int_I \bigr( u(t) \bigr)^4 dt. $$ This makes sense, because $H^1(I)\hookrightarrow C^0(I).$ I want to understand the Fréchet derivate $dF(u)[\phi]$ for $\phi\in H^1(I)$. To do so, I consider \begin{align} F(u+\phi)=\int_Iu^4 +4\int_I u^3\phi +6\int_I u^2\phi^2 + 4\int_I u\phi^3+\int_I\phi^4, \end{align} Clearly, the linear term should be $L_u(\phi):=4\int_I u^3\phi$; however when I want to prove it, I need to show that \begin{align} \lim_{\Vert\phi\Vert_{H^1}\to 0}\frac{\left\vert 6\int_I u^2\phi^2 + 4\int_I u\phi^3+\int_I\phi^4\right \vert}{\Vert \phi \Vert_{H^1}}=0. \end{align} However, I need to bound these integrals with the $H^1$ norm, but I have unsuccesful to do so. Are there any way to do it?
Thanks in advance.