This is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold,
$$\int e^w \sqrt{g}\, dx \leq C \exp \Big( c \|Du\|_{L^n}^n + \|u\|_{L^n}^n \Big),$$ where we used the standard Einstein notation.
I know I should use the Trudinger inequality, so I see that $$ |u(x)| = \frac{|u(x)|}{ \kappa \|D u\|_{L^n} } \cdot \kappa \|D u\|_{L^n}\leq \frac{n-1}{n} \left( \left( \frac{|u(x)|}{ \kappa \|DU\|_{L^n}} \right)^{\tfrac{n}{n-1}} + \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1}} \right).$$
So then
\begin{align*} \int e^{|u(x)|} dx &\leq \int \exp\left(\frac{n-1}{n} \left( \left( \frac{|u(x)|}{ \kappa \|DU\|_{L^n}} \right)^{\tfrac{n}{n-1}} + \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1}} \right)\right) dx \\ &\leq \int\left( \exp\left(\frac{n-1}{n} \left( \frac{|u(x)|}{ \kappa \|DU\|_{L^n}} \right)^{\tfrac{n}{n-1}} \right) \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1} }\right)\right) dx \\ & = \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1} }\right) \int \exp\left(\frac{n-1}{n} \left(\frac{|u(x)|}{ \kappa \|DU\|_{L^n}}\right)^{\tfrac{n}{n-1}}\right)\,dx \\ &\leq \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1} }\right) \cdot C \\ &= C \exp\left(\frac{n-1}{n} \Big( \kappa \| Du\|_{L^n} \Big)^{\tfrac{n}{n-1}}\right), \end{align*} since by the Trudinger inequality, $\displaystyle \int \exp\left(\frac{n-1}{n} \left(\frac{|u(x)|}{ \kappa \|DU\|_{L^n}}\right)^{\tfrac{n}{n-1}}\right) dx \leq C.$
Now, I would like to try to use partitions of unity to transport this inequality from $\mathbb{R}^n$ to manifolds, and also take advantage of cutoff functions. I read about them, but I'm not sure how to implement them. Do you think you could tell me how to do it?
Thank you very much in advance!