Current biased Josephson junctions are promising candidates for the detection of single photons in the microwave frequency domain. With modern fabrication technologies the switching properties of the junction can be adjusted to achieve quantum limited sensitivity. Namely, the width of the switching current distribution can be reduced well below the current amplitude produced by a single photon trapped inside a superconducting cavity. However for an effective detection a strong junction-cavity coupling is required, providing nonlinear system dynamics. We compare experimental findings for our prototype device with a theoretical analysis aimed to describe the switching dynamics of junctions under microwave irradiation. Measurements are found in qualitative agreement with our simulations.