Calculus of variations with delayed/shifted variable

Question

How can the following optimization problem be solved? $$ \min_{y} \quad \int_{-\infty}^{\infty} x^2y(x)\mathrm{d}x,\\ \mathrm{s.t.} \quad y(x)\leq \alpha y(x-d),\\ \int_{-\infty}^{\infty} y(x)\mathrm{d}x=1, $$ where $y$ is a real-valued function over $\mathbb{R}$. This is a problem of optimizing the second moment of a distribution with a non-classical constraint $y(x)\leq \alpha y(x-d)$ which makes it a bit ugly. Happy to hear what you think. Thanks.