Positive solutions to semipositone conjugate eigenvalue problems

✍ Scribed by Hua Su; Zhongli Wei

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 270 KB
• Volume: 69
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2007.09.012

✦ Synopsis

In this paper, we consider the existence of positive solutions to the following semipositone (k, nk) conjugate eigenvalue problems (SCEP):

where n ≥ 2, 1 < k < n -1 and λ > 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 semipositone and lower unbounded. Without making any monotone-type assumptions, by using the fixed-point index theory, we derive an explicit interval of λ such that for any λ 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 λ in an appropriate interval is also discussed.