Let $n$ be a positive integer greater than seven. Let $u\_a = (\frac{a}{a^2 + r^2})^\frac{n-4}{2}$, where $a$ is a positive real number. Let $\Delta u\_a$ be the Laplacian of $u\_a$. What is the dominant term of the small $a$ asymptotic expansion of the integral $\int\_{r=b}^\infty (\Delta u\_a)^2 r^{n-1} dr$, where $b$ is a small positive real number? 


This problem arises naturally when considering the convergence of Palais-Smale sequences of the Paneitz-Branson functional. I've tried using Maple to solve this problem, but it can't handle the problem due to the fact that $n$ is left undetermined. If $n$ is odd, I believe one can use the method of contour integrals in the complex plane to obtain a solution, so I'm primarily interested in the case where $n$ is even.



Asymptotics of a bubble
Actually, there is no need to do any change of variable. This is a straightforward calculation. Just use the binomial series to expand the integrand as an asymptotic series in negative powers of $r$.


