On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem $- \Delta u(x) = \lambda g(x) u(x)$ on $D$; $\frac{\partial u} {\partial n} (x) + \alpha u(x) = 0$ on $\partial D$, where $D$ is a bounded region in $\mathbf{R}^N$, $g$ is an indefinite weight function and $\alpha \in \mathbf{R}$ may be positive, negative or zero.