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How can I prove that the following Functional is Frechet Differentiable and that the Frechet derivative is continuous?
$$ I(u)=\int_\Omega |u|^{p+1} dx , \quad 1

where $\Omega$ is a bounded open subset of $\mathbb{R}^n$ and $I$ is a functional on $H^1_0(\Omega).$

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    1) compute the Gateaux derivative $\frac{d}{dt}I(u+t\psi)$; 2) check that it's continuous with respect to the basepoint $u$; 3) apply [this](http://math.stackexchange.com/questions/145910/gateaux-derivative/145993#145993).2012-08-13
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    @Leonid Kovalev, thanks, I tried that but I was stuck on Justifying the interchange between the limit and the integral in showing that $$ I^\prime(u_n)(\psi) \longrightarrow I^\prime(u)(\psi) $$ if $$ u_n\longrightarrow u $$2012-08-13
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    What did you get for $I'$?2012-08-14
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    @DavideGiraudo $$ I^\prime(u)(\psi)=(p+1)\int_\Omega |u|^{p-1}u\psi dx $$2012-08-14

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