Multiple positive solutions for a double phase system with singular nonlinearity

In this paper, we study a class of double phase systems which contain the singular and mixed nonlinear terms. Unlike the single equation, the mixed nonlinear terms make the problem more complicate. The geometry of the fibering mapping has multiple possibilities. To overcome the difficulties posed by the mixed nonlinear terms, we need to repeatedly construct concave functions, discuss different cases, and use the properties of concave functions and basic inequalities such as Holder inequality, Poincares inequality and Youngs inequality. By the use of the Nehari manifold, the existence and multiplicity of positive solutions which have nonnegative energy are obtained. It is worth mentioning that we note the existence of saddle point solution(a station point that is not a local minimum), see Remark 3.1.