Existence and nonexistence of entire solutions to the logistic differential equation

Vicentiu Radulescu,Marius Ghergu

doi:10.1155/s1085337503305020

Abstract

We consider the one-dimensional logistic problem(rαA(|u′|)u′)′=rαp(r)f(u)on(0,∞),u(0)>0,u′(0)=0, whereαis a positive constant andAis a continuous function such that the mappingtA(|t|)is increasing on(0,∞). The framework includes the case wherefandpare continuous and positive on(0,∞),f(0)=0, andfis 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 ofpandA. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.

Highlights

Introduction and the main resultsAs pointed out in the recent monograph by Buttazzo et al [3], one-dimensional variational problems deserve special attention
As we will see in this paper, higherdimensional variational problems can be reduced to one-dimensional ones
We study the following problem: rαA |u | u = rα p(r) f (u), r > 0, u(0) > 0, u (0) = 0, (1.1)

Summary

Introduction and the main results

As pointed out in the recent monograph by Buttazzo et al [3], one-dimensional variational problems deserve special attention. Our first result concerns the nonexistence of the solution to problem (1.1) in the case where limt→∞ tA(t) < ∞. The proof of Theorem 1.4 establishes that if problem (1.1) would have a solution necessarily, this solution blows up at infinity, that is, u(r) → +∞ as r → ∞. Such a solution is called explosive or large. Basic results in the study of large solutions for stationary problems have been recently obtained in [1, 2, 4, 5, 7, 8, 9, 10, 11]. If u is a positive solution of (1.1), lim r→∞

Proofs

Applications to the N-dimensional case