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

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