Discreteness of transmission eigenvalues for higher-order main terms and perturbations

In this paper we extend Sylvester's approach via upper triangular compact operators to establish the discreteness of transmission eigenvalues for higher-order main terms and higher-order perturbations. The coefficients of the perturbations must be sufficiently smooth and the coefficients of the higher-order terms of the perturbation must vanish in a neighborhood of the boundary of the underlying domain. The zeroth-order term must satisfy a suitable coercivity condition in a neighborhood of the boundary.