quantum mechanics - 2D Schrodinger equation in polar coordinates - boundary conditions at the origin

When solving the Schrodinger equation in 2D polar coordinates, one has to deal with various Bessel functions. In the most simple example, the infinite circular potential well, the solutions to the radial differential equation are the Bessel functions of first $[J_m(kr)]$ and second $[Y_m(kr)]$ kind. One usually discards the $Y_m(kr)$ functions on account of their asymptotic behavior at $r = 0$, $$Y_m(kr) \sim (kr)^{-m}$$ and so they are not square integrable functions. However, in the case of zero angular momentum, $m=0$, the Neumann function of zeroth order, $$Y_0(kr) \sim \ln (kr),$$ although infinite at the origin, is square integrable! So why do we have to discard it as well? What are the boundary conditions that have to be satisfied by a radial wave function at the origin?