Spectrum consisting in a continuous family plus an isolated point for a Dirichlet type problem

Eigenvalue problems involving the Laplace operator on bounded domains lead to a discrete or a continuous set of eigenvalues. In this paper we highlight the case of an eigenvalue problem involving the Laplace operator which possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, at least one more eigenvalue which is isolated in the set of eigenvalues of that problem. 2000 Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 15A18, 35P05.