On the positive solutions to some quasilinear elliptic partial differential equations

We establish that the elliptic equation ∆u + f(x, u) + g(|x|)x · ∇u = 0, where x ∈ R, n ≥ 3, and |x| > R > 0, has a positive solution which decays to 0 as |x| → +∞ under mild restrictions on the functions f, g. The main theorem extends and complements the conclusions of the recent paper [M. Ehrnström, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), 1147–1154]. Its proof relies on a general result about the long-time behavior of the logarithmic derivatives of solutions for a class of nonlinear ordinary differential equations and on the comparison method. Mathematics Subject Classification (2000). 34A12; 35J60.