On Lebesgue integration

Question

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?

Answer 1

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.