## The perturbation of and metric and standard bubble

Suppose ${\bar u_\varepsilon=\left(\frac{\varepsilon}{\varepsilon^2+|x|^2}\right)^{\frac{n-2}{2}}}$, then it is well know that it satisfies

$\displaystyle -\Delta \bar u_\varepsilon=n(n-2)\bar u^{\frac{n+2}{n-2}}_\varepsilon$

on ${\mathbb{R}^n}$. Now we want to perturb the Euclidean metric a little bit and an approximate solution of above equation. Suppose ${g(t)=\delta+tS+O(t^2)}$ where ${S=\frac{d}{dt}g(t)|_{t=0}}$ is a symmetric 2-tensor and ${\delta}$ is the Euclidean metric. The reasonable equation of ${u_\varepsilon=u_{\varepsilon}(t)}$ must satisfy should be

$\displaystyle -\Delta_{g}u_{\varepsilon}+\frac{n-2}{4(n-1)}R_gu_\varepsilon=n(n-2)u^{\frac{n+2}{n-2}}_\varepsilon$

Suppose ${u_\varepsilon(t)=\bar u_\varepsilon+tw+O(t^2)}$. Taking derivative on both sides of the above equation and restrict to ${t=0}$, one can get the equation ${w}$ must satisfy

$\displaystyle \Delta w+n(n+2)\bar u_\varepsilon^{\frac{4}{n-2}}w=\frac{n-2}{4(n-1)}\bar u_\varepsilon(\partial_i\partial_j S_{ij}-\Delta tr S)+\partial_{i}(\partial_j\bar u_\varepsilon S_{ij})-\partial_i(tr S)\partial_i\bar u_\varepsilon$

where we have use the fact that

$\displaystyle \frac{d}{dt}\big|_{t=0}\Delta_{g(t)}u_\varepsilon(t)=\Delta w-\partial_i(S_{ij}\partial_j\bar u_\varepsilon)+\frac 12\partial_i(tr S)\partial_i \bar u_\varepsilon$

$\displaystyle \frac{d}{dt}\big|_{t=0}R_{g(t)}u_\varepsilon(t)=(\partial_{i}\partial_jS_{ij}-\Delta tr S)\bar u_\varepsilon$