Estimates for eigenvalues of a class of fourth order degenerate elliptic operators with a singular potential

Abstract

In this paper, we consider estimates for eigenvalues of a class of fourth order degenerate elliptic operators with a singular potential. By using the Fourier transformation and some sub-elliptic estimates, we get the Li-Yau type lower bound estimates for the eigenvalues. Especially, if the degenerate vector fields are Grushin type vector fields satisfying the generalized Métivier condition, we not only obtain lower bound estimates, but also get the Yang-type inequalities as upper bounds for eigenvalues by using the trail function.