Degenerate elliptic eigenvalue problems with indefinite weights (Q839813)

This paper considers the following elliptic eigenvalue problem: \[ Au= \lambda m(x)u \quad\text{in } \Omega, \qquad Bu=0 \quad\text{on } \partial \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 2\), with smooth boundary \(\partial \Omega\), \[ Au= -\sum_{i=1}^N {\partial \over \partial x_i} \left(\sum_{j=1}^N a^{ij}(x) {\partial u\over \partial x_j}\right)+c(x)u \] is a second-order elliptic differential operator, and \(Bu=a(x'){\partial u\over \partial \nu}+b(x')u\) is the Robin boundary condition. The weight function \(m(x)\) may change sign and may be discontinuous. The purpose of the paper is to study the existence and uniqueness of solutions of the above eigenvalue problem in the framework of Sobolev spaces of \(L^p\) type. One main result is a theorem of the Krein and Rutman type, which assets that under appropriate conditions on \(a^{ij}(x)\), \(c(x)\), \(m(x)\), \(a(x')\) and \(b(x')\), the first eigenvalue \(\lambda_1(m)\) is positive and algebraically simple, with its corresponding eigenfunction \(\varphi_1(x)\in W^{2,p}(\Omega)\), \(N<p<\infty\), which may be chosen to be strictly positive in \(\Omega\).