If $u, v \in H_0^1(\Omega)$ then $|u|^{p-1}|v| \in L^1$

Question

Why is it that if $u, v \in H_0^1(\Omega)$ then $|u|^{p-1}|v| \in L^1$?

The context of this question is the computation of the Gateaux derivative of the functional $$ J(u) = \int_\Omega F(x, u(x))dx $$ where $F(x, t) = \int_0^t f(x, s)ds $ and $f$ is such that there are constants $a, b > 0$ such that $$ |f(x, t)| \leq a + b|t|^{p-1}, $$ for $ 1 < p \leq 2^*$. Here, $\Omega$ is a smooth domain of $\Bbb{R}^N$, $N \geq 3$.

Answer 1

With the revised assumptions the result is correct: Recall the generalized form of Holder's inequality that says if $1/r=1/p+1/q$ then $$ \| fg\|_r\leq \|f\|_p\| g\|_q. $$ Use this with $f=|u|^{p-1}$ and $g=v$ and $q=2^*=2n/(n-2)$, $p=2_*=2n/(n+2)$.

It's easy to check that $|u|^{p-1}\in L^{2_*}$ if and only if $p\leq 2^*$.