Positive solutions to a singular differential equation of second order

This paper concerns the existence of positive solutions to a singular differential equation u″+λu′t−γ|u′|2u+f(t)=0,00λ,γ>0, f(t)∈C[0,1]f(t)∈C[0,1] and f(t)>0f(t)>0 on [0,1][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−1λ=N−1 (NN is a positive integer) and f≡1f≡1, the problem admits a decreasing positive solution. In this paper, we show, by the classical method of elliptical regularization, that if γ>12(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−1λ=N−1 and f≡1f≡1 admits at least two positive solutions if γ>N2.