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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...

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Integration with weight $1-s$ gives you the second antiderivative, i.e., $\int_0^1\int_0^t |\phi_1+s\phi|^{p-2}\phi^2 \,ds\,dt$. Maybe the integrand is related to the second derivative of something? –  user53153 Feb 14 '13 at 20:44
    
Good idea, I think that the second derivative of $G$ will do the job. –  Tomás Feb 14 '13 at 22:39
    
Yes Pavel, the second derivative solves the problem: $G''(\phi_1+s\phi)\phi\phi=(p-1)\int |\phi_1+s\phi|^{p-2}\phi^2$. By using integration by parts the problem is solved. Thank you –  Tomás Feb 15 '13 at 12:21

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