I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page.
Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields
$$
\int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$
where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$?
When $\frac{2^*}{3}>1$ then using Holder inequality and Sobolev inequality we can bound the term $\int_{\mathbb{R}^n}\sigma^{p-2}\rho^3$ by $O(\int_{\mathbb{R}^n}{|\nabla\rho|^2})^{\frac{3}{2}}$ but I am not getting the other exponent of $\gamma$ i.e $\frac{2}{n-2}$ and the minimum over both the two would be the exponent. Any help is very much appreciated.
