Let $Y$ be a random exponentially distributed variable, with mean $\lambda$. That is, its probability density function is $$f(y) = \left\{ \begin{array}{ c c } \frac{1}{\lambda}e^{\frac{-y}{\lambda}} & y > 0 \\ 0 & y \leq 0 \end{array} \right.$$
I want to perform a likelihood ratio test to test $$\left\{\begin{array}{ c c c } H_0 & : & \lambda \leq \lambda_0 \\ H_1 & : & \lambda > \lambda_0 \end{array} \right.$$
I derive the likelihood $$L(\lambda \mid y) = \left(\frac{1}{\lambda}\right)^n \exp\left(\frac{n\overline{y}}{\lambda}\right)$$ as well as the log-likelihood $$l(\lambda \mid y) = n \log\left(\frac{1}{\lambda}\right) - \frac{n\overline{y}}{\lambda}$$
I also know the MLE for $\lambda$, $$\hat{\lambda} = \overline{y}$$
I have read this question as well as this one, but just do not understand how to proceed now.