Abstract and Applied Analysis
Volume 2003 (2003), Issue 17, Pages 995-1003
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 Marius Ghergu and Vicenţiu Rădulescu. 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.
We consider the one-dimensional logistic problem
, , where is a positive constant and
is a continuous function such that the mapping is increasing on . The framework includes the case where
and are continuous and positive on , , and 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 and . As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.