Positive solutions to a singular differential equation of second order

✍ Scribed by Wenshu Zhou; Shoufeng Cai

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 222 KB
• Volume: 68
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2007.01.048

✦ Synopsis

This paper concerns the existence of positive solutions to a singular differential equation

with the boundary conditions

where λ, γ > 0, f (t) ∈ C[0, 1] and f (t) > 0 on [0, 1]. By theories of ordinary differential equations, Bertsch and Ughi [M. Bertsch, M. Ughi, Positivity properties of viscosity solutions of a degenerate parabolic equation, Nonlinear Anal. 14 (1990) 571-592] proved that when λ = N -1 (N is a positive integer) and f ≡ 1, the problem admits a decreasing positive solution. In this paper, we show, by the classical method of elliptical regularization, that if γ > 1 2 (1 + λ), then the above problem admits at least a positive solution which is not decreasing. As a by-product of the results, the problem with λ = N -1 and f ≡ 1 admits at least two positive solutions if γ > N 2 .