Calculus of Variations: Finding the Unit-Norm Filter With the Maximum Response

Question

I'm trying to solve the following problem: given a smooth function $f:\mathbb{R}\to\mathbb{R}$, find the filter $\phi:[-\tau,\tau] \to \mathbb{R}$ that maximizes the functional \begin{align*} I[\phi] = \int_0^T \bigg(\int_{-\tau}^{\tau}\phi(p)f(t+p) \,dp \bigg)^2 \,dt \end{align*} subject to the constraint that \begin{align*} \int_{-\tau}^{\tau} \phi(p)^2\,dp = 1 \end{align*} To solve this, I want to apply the method of Lagrange multipliers. I first stationarized the auxiliary functional \begin{align*} \int_0^T \bigg(\int_{-\tau}^{\tau}\phi(p)f(t+p) \,dp \bigg)^2 \,dt - \lambda \bigg(\int_{-\tau}^{\tau} \phi(p)^2\,dp - 1\bigg) \end{align*} From this I obtained the following equation: \begin{align*} \int_0^T x(t+p) \int_{-\tau}^\tau \phi(q) x(t + q)\,dq\,dt - \lambda \phi(p) = 0 \end{align*} It's at this point that I'm unsure of how to proceed. I've seen in some other examples of using the Lagrange multiplier method that the next step should be to solve for $\lambda$ by combining the constraint equation and the variational equation. However, I'm not sure how to do this. Can anybody help me solve this problem?