Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\phi_1,v,\phi\in W_0^{1,p}(\Omega)$ with $p\in (1,\infty)$. How can I evaluate the integral: $$\int_0^1F(s)ds$$ where $F(s)=\int_\Omega|\phi_1+s\phi|^{p-2}\phi^2(1-s)$
Edit: If we define $G:W_0^{1,p}(\Omega)\rightarrow\mathbb{R}$ by $$G(u)=\frac{1}{p}\int_\Omega|u|^p$$
we have that $$G'(\phi_1+s\phi)\phi=\int_\Omega|\phi_1+s\phi|^{p-2}(\phi_1+s\phi)\phi$$
Maybe the last equatily can help in some way...