Existence and nonexistence of entire solutions to the logistic differential equation, Abstract and Applied Analysis 17

We consider the one-dimensional logistic problem (rαA(|u′|)u′) ′ = rαp(r) f (u) on (0,∞), u(0)> 0, u′(0) = 0, where α is a positive constant and A is a continu-ous function such that the mapping tA(|t|) is increasing on (0,∞). The frame-work includes the case where f and p are continuous and positive on (0,∞), f (0) = 0, and f is nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results de-pending on the growth of p and A. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative so-lutions, excepting the trivial one