Regular eigenvalue problem with eigenparameter contained in the equation and the boundary conditions (Q803411)

For a bounded region \(\Omega \subset {\mathbb{R}}^ n\) with smooth boundary, the following eigenvalue problem is considered: \(-(1/r)\Delta_ nu=\lambda u\text{ in } \Omega,\quad \partial u/\partial \gamma +h(x)u=\lambda u\text{ on } \partial \Omega.\) Here, \(\Delta_ n\) denotes the Laplacian, r a positive weight function satisfying some Hölder condition, \(\lambda\) the eigenvalue parameter, and h a non-negative function on \(\partial \Omega\) which may be discontinuous. Introducing an appropriate Hilbert space, the problem is reduced to an operator equation \(Au=\lambda u\), where the linear operator A is symmetric, strictly positive and unbounded from above. It is shown: The spectrum consists of real eigenvalues \(0<\lambda_ 1\leq \lambda_ 2\leq..\)., \(\lambda_ n\to \infty\) as \(n\to \infty\), and the corresponding eigenfunctions form a complete system in the Hilbert space.