Báo Cáo Báo cáo hóa học THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS

THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS D. A. KANDILAKIS, M. MAGIROPOULOS, AND N. B. ZOGRAPHOPOULOS Received 12 October 2004 and in revised form 21 January 2005 We study the properties of the positive principal eigenvalue and the corresponding eigenspaces of two quasilinear elliptic systems under nonlinear boundary conditions. We prove that this eigenvalue is simple, unique up to positive eigenfunctions for both systems, and isolated for one of them. 1. Introduction Let Ω be an unbounded domain in RN , N ≥ 2, with a noncompact and smooth boundary ∂Ω. In this paper we prove certain properties of the principal eigenvalue of the following quasilinear elliptic systems ư∆ p u = λa(x)|u| pư2 u + λb(x)|u|αư1 |v |β+1 u, ư∆q v = λd(x)|v |qư2 v + λb(x)|u|α+1 |v |βư1 v, ư∆ p u = λa(x)|u| pư2 u + λb(x)|u|α |v |β v ư∆q v = λd(x)|v |qư2 v + λb(x)|u|α |v |β u in Ω, in Ω, in Ω, in complex plane C. We denote by D+ and Dư the bounded and unbounded