Is there a UMP test at level $\alpha$?

Question

Let $Y_1$, . . . , $Y_n$ be a random sample from a distribution with the density function:

$f_θ(y) = \frac {3θ^3}{y^4}$ for y ≥ $\theta$ > 0

Is there a UMP test at level $\alpha$ for testing $H_0$ : $\theta$ ≤ $θ_0$ vs. $H_1$ : $\theta$ > $\theta_0$?

My initial thoughts are to look for an MLR, so the likelihood ratio is as such:

$\frac {L(\theta_2 ; y)}{L(\theta_1 ; y)}$ = $\frac {\theta_2^{3n}} {\theta_1^{3n}}$

This is a non-decreasing function, but it isn't a function of a statistic, so there is no MLR. I get stuck after here, because if there is no MLR does that mean that there is definitely not a UMP test, or do I need to search for it another way?