  In this paper, we firstly consider Dirichlet eigenvalue problem which is
related to Xin-Laplacian on the bounded domain of complete Riemannian
manifolds. By establishing the general formulas, combining with some results of
Chen and Cheng type, we prove some eigenvalue inequalities. As some
applications, we consider the eigenvalues on some Riemannian manifolds
admitting with special functions, the translating solitons, minimal
submanifolds on the Euclidean spaces, submanifolds on the unit spheres,
projective spaces and so on. In particular, for the case of translating
solitons, some eigenvalue inequalities are universal. Moreover, we investigate
the closed eigenvalue problem for the Xin-Laplacian and generalize the Reilly's
result on the first eigenvalue of the Laplace-Beltrami operator. As some
remarkable applications, we obtain a very sharp estimate for the upper bound of
the second nonzero eigenvalue(without counting multiplicities of eigenvalues)
of the Laplace-Beltrami operator on the minimal isoparametric hypersurfaces and
focal submanifolds in the unit sphere, which leads to a conjecture and is the
most fascinating part of this paper.
