I've been reading papers about a fairly unknown topic in quantum mechanics called the quantum backflow effect. And in many of the papers they find an eigen value problem corresponding to the maximal amount of backflow. I understand all of the proof except from the last step in which they find the Euler-Lagrange of a functional ($I(\phi)$) to maximise it.
$$I(\phi) = \int^\infty_0{\int^\infty_0{\phi^*(p)K(p,q)\phi(q)dp}dq} - \lambda\int^\infty_0{\phi^*(p)\phi(p)dp}$$
Is the functional to be maximised and the Euler-Lagrange is
$$\int^\infty_0{K(p,q)\phi(q)dq} = \lambda\phi(p).$$
Although none of the papers explain why this is the case and I've been searching the internet for how to solve a problem in this form for days and can't seem to find anything. If anyone knows why this is the Euler-Lagrange of $I(\phi)$ any help would be greatly appreciated.