How to prove that this function $$|\phi(x)|\left(2a+b(|u(x)|+|\phi(x)|)^{p-1}+b|u(x)|^{p-1}\right)$$ is in $L^{1}(\Omega)$, where $u,\phi\in L^{p}(\Omega)$,$\Omega \subset \Bbb{R^{n}} $ be a bounded open set, a,b are constants?
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I've edited your question because using ${,}$ as delimiters tends to be confusing. Let me know if I've introduced an error. Something seems wrong here-should the second exponent also be outside some parentheses? – Kevin Carlson Jul 31 '13 at 07:10
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Yeah, this isn't true in current form: take $\Omega=[0,1],\phi=1,u=1/x,p=2,a=0,b=1$. $1+2/x$ is not in $L^1[0,1]$. – Kevin Carlson Jul 31 '13 at 07:16
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@kevin-carlson i cant able to understand is i did any mistake in my question – nanthini Jul 31 '13 at 07:23
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I do think there's a mistake in your question. – Kevin Carlson Jul 31 '13 at 07:27
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What is $\Omega$? – Davide Giraudo Jul 31 '13 at 10:05
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@davide-giraudo $\Omega \subset \Bbb{R^{n}}$ be a bounded open set – nanthini Jul 31 '13 at 10:17
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I assume $p>1$. Since $(x+y)^p\leqslant 2^{p-1}(|x|^p+|y|^p)$ and $L^p(\Omega)\subset L^1(\Omega)$, we only have to prove that $|\phi|\cdot |u|^{p-1}$ is integrable. This is a consequence of Hölder's inequality.
Davide Giraudo
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