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