Cognitive radio and non-orthogonal multiple access are promising for alleviating the severe spectral scarcity problem encountered by the next generation wireless communication systems. In this paper, in order to improve energy efficiency and spectral efficiency, a non-orthogonal multiple access cognitive radio network with simultaneous wireless information and power transfer is studied under a practical non-linear energy harvesting model. A multi-objective resource optimization problem is formulated for maximizing the harvesting power of each energy harvesting receiver. This problem is non-convex and challenging to solve. A weighted Tchebycheff method is applied to solve the formulated problem. It is shown that the performance achieved under the non-linear energy harvesting model is better than that obtained under the linear energy harvesting model.