I am wondering if the solution of Laplace's equation (e.g. electrostatic potential $\Phi$ which satisfies $\nabla^2\Phi=0$) is square-integrable? I am confused, because in one dimension in the large distance limit, $\Phi(x)\sim\frac{1}{x}$. So $\int_{-\infty}^{\infty}\Phi^2\ dx<\infty$. However, if I go to the large distance limit in the three dimensions, $\int_{r=0}^{\infty}\int_ {\theta=0}^{\pi}\int_{\phi=0}^{2\pi}\Phi(r)^2\ d^3{\bf r}$. This seems to diverge, even though $\Phi\sim\frac{1}{r}$. It is true that the higher order multipole terms converge, but the monopole term still seems to dominate.
For me it looks odd that such a basic principle should exhibit different behaviour in one and three dimensions. Can anyone kindly explain if I am going wrong?
Regards, Kolahal