Edge-flames have complex structures with lean and/or rich premixed branches and an elongated diffusion flame. Such structures are seen at the base of lifted diffusion flames, and their dynamical properties have an important effect on flame stabilization, onset of flame oscillations and/or blowout. They are also observed near holes created in turbulent diffusion flames, and their dynamical behavior is responsible to either spreading extinction to other parts of the flame surface or reestablishing burning in the hole region. Previous theoretical studies of edge-flames were all based on constant-density models, which practically assume that the flame does not have any effect on the underlying flow. This work marks the first attempt to address what effect does thermal expansion have on flame dynamics. We find, similar to the earlier predictions, two modes of flame stabilization: a steady mode at low injection velocities and an oscillatory mode at higher velocities. The gas expansion has an effect on the flame standoff distance: as a result of the reduced density in the preheat zone the flow accelerates when crossing the flame which consequently forces its edge to relocate at an upstream position where its propagation speed balances the gas velocity. The onset of oscillations at relatively high flow rates is predicted with or without invoking the constant-density approximation; the critical conditions, however, are affected by density variations.