Im a math major student and taking a course in multivariable calculus.
I straggled with the following homework exercise.
Let $u,v :\mathbb{R}^n \to \mathbb{R}$ be harmonic functions (i.e. $\Delta u,\Delta v \equiv 0)$, and let $a,b,p,q>0$ constants such that $ab=pq$.
Prove that: $$ \int_{S^{n-1}}u(ax)v(bx)-u(px)v(qx)\,\mathrm dS(x)=0, $$ where $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$.
As a context for the problem, I'll add that the main tools we have been taught before this exercise (concerning harnomic functions) are: The average principle, the maximum principle, Liouville's theorem and Poisson kernel
My first attempt was to check if the integrand is harmonic but it not seems to be the case. After that I thought to add some function under the integral so the new integrand will become simpler, but I fell short on that attempt too.
Can you guys give me direction or a hint?