  We study the emission from a molecular photonic cavity formed by two proximal
photonic crystal defect cavities containing a small number (<3) of In(Ga)As
quantum dots. Under strong excitation we observe photoluminescence from the
bonding and antibonding modes in excellent agreement with expectations from
numerical simulations. Power dependent measurements reveal an unexpected peak,
emerging at an energy between the bonding and antibonding modes of the
molecule. Temperature dependent measurements show that this unexpected feature
is photonic in origin. Time-resolved measurements show the emergent peak
exhibits a lifetime $\tau_M=0.75 \, \pm 0.1 \, ns $, similar to both bonding
and antibonding coupled modes. Comparison of experimental results with
theoretical expectations reveal that this new feature arises from a coexistence
of weak- and strong-coupling, due to the molecule emitting in an environment
whose configuration permits or, on the contrary, impedes its strong-coupling.
This scenario is reproduced theoretically for our particular geometry with a
master equation reduced to the key ingredients of its dynamics. Excellent
qualitative agreement is obtained between experiment and theory, showing how
solid-state cavity QED can reveal new regimes of light-matter interaction.
