I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: [arXiv link](https://arxiv.org/abs/1602.01916), [link at Figalli's page](https://people.math.ethz.ch/~afigalli/papers-pdf/A-quantitative-analysis-of-metrics-on-Rn-with-almost-constant-positive-scalar-curvature-with-applications-to-fast-diffusion-flows.pdf).

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 $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."<br>
 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. 


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