The paper concerns sandwich panels that consist of two thin but stiff facings and a thick but deformable core. These types of structures is widely used in the aerospace, shipbuilding and automotive industries, as well as in civil engineering. Despite their numerous advantages, sandwich structures are characterized by significant sensitivity to local instability of thin facings. This study considers the case in which the panel core is made of a physically non-linear material. The nonlinear behavior of the core is due to its cellular structure, which responds differently to compression and tension. The problem of determining the critical stress was solved using the energy method, and the obtained solution was verified using numerical simulation. The obtained solution was extended to various cases of boundary conditions that are close to the actual support conditions of the facing resting on the deformable core.