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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.

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Do I understand correctly that $r$ in the definition of $u_a$ is notation for $\sqrt{x_1^2\dots+x_n^2}$ and Laplacian is taken in $\mathbb{R}^n$? – Fedor Petrov Aug 20 '11 at 8:03
Judging from the $n>7$ condition, we are in $\mathbb R^3$ but I'd prefer the OP to confirm this before attempting any computation. – fedja Aug 20 '11 at 12:57
Do a change of variable: $r = as$. – Deane Yang Aug 20 '11 at 13:06
And take more applied math or physics courses. They're really useful for learning how to do calculations. – Deane Yang Aug 20 '11 at 13:11
@Fedor Petrov: You are correct. @fedja: We are actually working in $R^n$. – Viktor Bundle Aug 20 '11 at 16:12
up vote 1 down vote accepted

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 $r$.

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