Positive solutions to semipositone (k, n-k) conjugate eigenvalue problems

摘要

In this paper, we consider the existence of positive solutions to the following semipositone (k, n - k) conjugate eigenvalue problems (SCEP): {(-1)((n-k))u((n)) (t) = lambda f(t, u(t)). 0 < t < 1. u((i))(0) = 0. 0 <= i <= k - 1. u((j))(1) = 0. 0 <= j <= n - k -k. where n >= 2.1 < k < n - 1 and lambda > 0 is positive parameter. The nonlinear function f may change sign for 0 < t < 1, i.e., we allow that the nonlinear term f is both semipositione and lower unbounded. Without making any monotone-type assumptions, by using the fixed-point index theory, we derive an explicit interval of lambda such that for any lambda in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed, and the existence of at least two solutions for lambda in an appropriate interval is also discussed (C) 2007 Elsevier Ltd. All rights reserved.