PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 1, January 1999, Pages 125–130 S 0002-9939(99)04561-X ON PRINCIPAL EIGENVALUES FOR BOUNDARY VALUE PROBLEMS WITH INDEFINITE WEIGHT AND ROBIN BOUNDARY CONDITIONS G. A. AFROUZI AND K. J. BROWN (Communicated by Jeffrey B. Rauch) Abstract. We investigate the existence of principal eigenvalues (i.e., eigenval- ues corresponding to positive eigenfunctions) for the boundary value problem -Δu(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 problem −Δu(x)= λg(x)u(x) on D; ∂u ∂n (x)+ αu(x) = 0 on ∂D, (1) α where 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 Bˆ ocher (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) α ...λ − 2 <λ − 1 < 0 <λ + 1 <λ + 2 ..., λ + 1 (λ − 1 ) being the unique positive (negative) principal eigenvalue. It is also well known that the case where 0 <α< ∞ is similar to the Dirichlet case. In the case of Neumann boundary conditions, 0 is clearly a principal eigenvalue and there is a positive (negative) principal eigenvalue if and only if  D g(x) dx < 0(> 0); in the case where  D g(x) = 0 there are no positive and no negative principal eigenvalues. 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 Received by the editors April 30, 1997. 1991 Mathematics Subject Classification. Primary 35J15, 35J25. Key words and phrases. Indefinite weight function, principal eigenvalues. The first author gratefully acknowledges financial support from the Ministry of Culture and Higher Education of the Iran Islamic Republic. c 1999 American Mathematical Society 125 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use