Define $J\colon W\_0^{1,2}(\Omega)\to \mathbb{R}$ by $J(u)=\int\_\Omega u^{p+1}dx$ for $p\in (1,\frac{n+2}{n-2})$.


Is $J'(u)(v)=\int\_\Omega (p+1)u^pvdx$?



Derivative of functional
Yes.  

Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get 
$$\bigg|\, J(u+v)-J(u)-(p+1)\int\_\Omega |u|^p v\, dx\, \bigg| \le p(p+1)\int\_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \, dx.$$ 
Use the Hoelder inequality to check that the RHS is actually $o(\|v\|\_{p+1}).$ You can also use he dominated convergence theorem to show that $J$ is $C^1$. Your Sobolev space case follows immediately by composition with the Gagliardo-Nirenberg-Sobolev embedding (assuming you have $\Omega$ bounded in the case $p+1\neq 2^\*$, otherwise $J(u)$ may be not even defined for some $u$). 


For more general facts on integral functionals and Nemytskii operators go and see the great texts, Veinberger, Krasnoselskii, Deimling &c. And if you look for a nice, friendly introductory book to Nonlinear Analysis, with beautiful theory and clever applications, I'd suggest [*A Primer of Nonlinear Analysis*](http://books.google.it/books?id=dVSTUben3QoC&printsec=frontcover&dq=ambrosetti+prodi+a+primer&source=bl&ots=MuTPItXqIe&sig=FenOfPhmYQIDYKwMcedIpgcy194&hl=it&ei=GKIcTbrjH8eOswbu6p2qAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBoQ6AEwAA#v=onepage&q&f=false) by Antonio Ambrosetti and Giovanni Prodi (resp. my mathematical father and grandfather ;-).


