Resently new inhomogeneous field configurations similar to kinks in the Gross-Neveu model were found in the CP(N-1) model in the large N limit. We will discuss properties of the solutions. In particular, energy of the inhomogeneous states is lower then energy of the homogeneous one. Thus the question about the true ground state of the model arises. In context of this question we discuss periodic elliptic solutions. The talk is mainly based on the preprint (arXiv:1811.05449).