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Abstract and Applied Analysis
Volume 2003 (2003), Issue 17, Pages 995-1003
http://dx.doi.org/10.1155/S1085337503305020

Existence and nonexistence of entire solutions to the logistic differential equation

Department of Mathematics, University of Craiova, Craiova 200 585, Romania

Received 26 February 2003

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 continuous function such that the mapping tA(|t|) is increasing on (0,). The framework 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 depending 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 solutions, excepting the trivial one.