Question

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

Answer 1

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 by Antonio Ambrosetti and Giovanni Prodi (resp. my mathematical father and grandfather ;-).