On the asymptotic behavior of the principal eigenvalues of some elliptic problems

This paper is concerned with nonself-adjoint elliptic problems involving indefinite weights and boundary conditions of the Dirichlet, Neumann or Robin type. We study the asymptotic behavior of the principal eigenvalues, when the first order term (drift term) becomes larger and larger. The basic results of Berestycki et al. (Commun. Math. Phys., 253:451-480, 2005) are extended to the present context. Moreover, answers are provided to some open problems raised in Berestycki et al. (Commun. Math. Phys., 253:451-480, 2005).