Positive solutions to a singular differential equation of second order

This paper concerns the existence of positive Solutions to a singular differential equation u '' + lambda u'/t - gamma|u'|(2)/u + f(t) = 0, 0 < t < 1, with the boundary conditions u(1) = u'(0) = 0, where lambda, gamma > 0, f(t) is an element of 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 de-generate parabolic equation, Nonlinear Anal. 14 (1990) 571-592] proved that when lambda = N - 1 (N is a positive integer) and f equivalent to 1, the problem admits a decreasing positive solution. In this paper, we show, by the classical method of elliptical regularization, that if gamma > 1/2 (1 + lambda), then the above problem admits atleast a positive solution which is not decreasing. As a by-product of the results, the problem with lambda = N - 1 and f equivalent to 1 admits atleast two positive solutions if gamma > N/2. (C) 2007 Elsevier Ltd. All rights reserved.