Orbital Stability of Optical Solitons in 2d

We present a stability result for ground states of a Schr\"odinger-Poisson system in $(2+1)$ dimension, modelling the propagation of a light beam through a liquid crystal with nonlocal nonlinear response. The core of the proof is a coercivity bound on the second derivative of the action, where non scaling nonlinearities and the coupled system present the major difficulties. In addition we prove existence of a ground state with frequency $\sigma$ for any $\sigma \in (0,1)$ as a minimal point over an appropriate Nehari manifold.