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

Abstract

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem -Au(x) = Ag(x)u(x) onD; , (x) +au(x) 0on D a (x) + au(x) = 0 on AD, where D is a bounded region in RN with smooth boundary, g: D -R is a smooth function which changes sign on D, and a E R. Such problems have been studied in recent years because of associated nonlinear problems arising in the study of population genetics (see [3]). The study of the linear ordinary differential equation case, however, goes back to Picone and Bocher (see [2]). Attention has been confined mainly to the cases of Dirichlet (a = oc) and Neumann boundary conditions. In the case of Dirichlet boundary conditions it is well known (see [4]) that there exists a double sequence of eigenvalues for (1)> * **\> 0); in the case where fD g(x) = 0 there are no positive and no negative principal eigenvalues. We shall investigate how the principal eigenvalues of (1), depend on a, obtaining new results for the case where a 0, probably because it is more natural that the flux across the boundary should be outwards if there is a positive concentration at the Received by the editors April 30, 1997. 1991 Mathematics Subject Classification. Primary 35J15, 35J25.