Applying Holder Inequality to obtain a estimate.

Question

I'm reading a paper concerned with PDE, a inequality in it makes me confuse. We have known: $$\int|\nabla u|^{p-2}|A|^2\eta^2\leq C\int|\nabla u|^{p}$$ and author says we can obtain following inequality by applying Holder inequality: $$\int|\nabla u|^{p-2}|A|\eta^2\leq C\left(\int|\nabla u|^{p}\right)^{\frac{p-1}{p}}$$ in which $p\geq 2$. I believe that to applying the first inequality, we need all of three terms square, but that can't achieve the form we want. Any hint are very welcomed and appreciated.

Answer 1

I believe that I have found the correct exponent, it's

$$\int|\nabla u|^{p-2}|A|\eta^2\leq\left(\int|\nabla u|^{p-2}|A|^2\eta^2\right)^{\frac{1}{2}}\left(\int|\nabla u|^{p}\right)^{\frac{p-1}{p}-\frac{1}{2}}\left(\int\eta^p\right)^{\frac{1}{p}}\leq C(\int|\nabla u|^{p})^{\frac{p-1}{p}}$$

I will shut down the problem, thanks for comments.