Eigenvalue criteria for existence of positive solutions Of second-order, multi-point, p-laplacian boundary value problems

In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem (1) - (Ø_p(u?)?+(a+g(x,u,u?)Ø_p(u) =0, a.e. on (-1,1), (2) u(±1)=S_i=1 ^±a_i^±u(? _i^±) where p>1, Ø_p(s) ^p-2s, se R,m± = 1 are integer, and ?_i^±?(-1,1), a_i^± > 0, i=1 ,?,m^±, S_i=1^m±a_i^± Also a ? L¹(-1,1) and g: [-1,1]× R ²?R is Carath´eodory, with (3) g(x,0,0)=0, x ?[-1,1]. Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of g(x, s, t), as s ? 8, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation (4) -Ø(u?)?+aØp(u)= ?Øp(u), on (-1,1) Copyright © 2010 Juliusz Schauder Center for Nonlinear Studies.