What is the fundamental solution of the linear operator $L = \nabla^2 u - k^2 u$ on $\mathbb{R}^2$, with the constraint that the solution goes to zero at infinity? I've figured out that $u(r) = aK_0(kr)$, where $K_0$ is the zeroth modified Bessel function, but cannot find $a$.
I understand that what I need to do is integrate the equation on both sides, but I cannot estimate how $LK_0(kr)$ behaves near the origin!