Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let be a bounded domain with smooth boundary in an n-dimensional metric measure space and let be a vector-valued function from to . In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: , in , and where is the drifting Laplacian and is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space and the Gaussian shrinking soliton . Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton , where is an Einstein manifold with constant Ricci curvature .