Abstract. We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem −∆u(x) = λg(x)u(x) on D; ∂u ∂n (x)+ αu(x) = 0 on ∂D, where D is a bounded region in R N , g is an indefinite weight function and α ∈ R may be positive, negative or zero.We discuss the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problemwhere D is a bounded region in R N with smooth boundary, g : D → R is a smooth function which changes sign on D, and α ∈ 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 Bôcher (see [2]). Attention has been confined mainly to the cases of Dirichlet (α = ∞) 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) We shall investigate how the principal eigenvalues of (1) α depend on α, obtaining new results for the case where α < 0. This case seems to have been considered far less often than the case α ≥ 0, probably because it is more natural that the flux across the boundary should be outwards if there is a positive concentration at the

We study a reaction᎐diffusion version on all of ‫ޒ‬ N of the logistic equation of population growth in which the birth rate depends on the spatial variable and may assume both positive and negative values. Our results which are obtained by the construction of sub-and supersolutions and the study of asymptotic properties of solutions show the interplay between the birth rate of the species and the extent of diffusion in determining the existence or nonexistence of nontrivial steady-state distributions of population. ᮊ

We study the principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem:where ∆ is the standard Laplace operator, D is a bounded domain with smooth boundary, g : D → R is a smooth function which changes sign on D and α ∈ R. We discuss the relation between α and the principal eigenvalues.2000 Mathematics Subject Classification: 35J15, 35J25.

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