EXISTENCE AND NONEXISTENCE OF ENTIRE SOLUTIONS TO THE LOGISTIC DIFFERENTIAL EQUATION

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.


Introduction and the main results
As pointed out in the recent monograph by Buttazzo et al. [3], one-dimensional variational problems deserve special attention.In fact, problems of this kind have their own characters.Sometimes, as we will also see in this paper, higherdimensional variational problems can be reduced to one-dimensional ones.
in the radial case.
We point out that the case A(t) = t m−2 , m > 2, was studied in [11].
then (1.1) has no positive solutions.
In fact, the proof of Theorem 1.4 establishes that if problem (1.1) would have a solution then, necessarily, this solution blows up at infinity, that is, u(r) → +∞ as r → ∞.Such a solution is called explosive or large.There is a great interest in the last few decades regarding the study of solutions that blow up at the boundary or at infinity.If m = 2, then condition (1.8) is known as the Keller-Osserman condition (see [6,13]) and it plays a basic role in the treatment of elliptic equation that admits large solutions.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].
then (1.1) has at least one positive solution.Moreover, this solution is large.
Our next result gives an estimate of the growth of a solution of (1.1) in case if f is bounded.More precisely, we prove the following theorem.

Proofs
If u is a positive solution of (1.1), then We deduce that A(|u (r)|)u (r) > 0 for r > 0 which implies u (r) > 0. Since f is nondecreasing, it follows that

1) and (2.3) yield
that is, Proof of Theorem 1.2.Arguing by contradiction, let u be a solution of (1.1).Since u(0) > 0 and f , u are nondecreasing functions, from (2.1) we get On the other hand, lim t→∞ tA(t) < ∞, which implies that A(u (r))u (r) is bounded on [0,∞).This fact and the above inequality lead to a contradiction since g(r) → ∞ as r → ∞ and f (u(0)) > 0. The proof of Theorem 1.2 is now complete.
Proof of Theorem 1.4.Assume by contradiction that problem (1.1) has a positive solution u.From (2.6), we get u (r) → ∞ as r → ∞ and so, by Remark 1.3, u(r) → ∞ as r → ∞.By (g2) and (2.5), we have Now, integration by parts yields for all r > 0. By a change of variables, we now find By (1.7) and using the fact that u (r) → +∞ as r → ∞, there exist r 0 > 0 and a positive constant C > 0 such that (2.12) M. Ghergu and V. Rȃdulescu 999 Hence, Integrating this inequality on [r 0 ,r], we find Letting r → ∞ in the above relation, we get This contradicts our assumption (1.8) and completes the proof.
Proof of Theorem 1.5.The existence of a solution u of (1.1) in a certain interval [0,R) follows by the classical arguments of ODEs.Assume, by contradiction, that the maximal interval of existence of 1), we deduce that u (R − 0) exists and is finite.Then, by standard arguments for initial value problems, it follows that u can be extended as a solution on an interval [0,R + ε), ε > 0, which contradicts the maximality of R. Hence, u(R − 0) = ∞.
Using (2.1) and the fact that A(u )u ≥ 0 on [0,R), we have where C 0 = max r∈[0,R] g(r) > 0. Multiplying the above inequality by u ≥ 0 and integrating on [0,r], we have (2.17) According to (1.7), there exists R 0 ∈ (0,R) such that where C 1 > 0 is a constant independent of f and u.Hence, An integration over [R 0 ,r], r < R, and a change of variable lead to which contradicts our assumption (1.9).We conclude that there exists a solution of (1.1) and the proof is now complete.
A very useful tool is the following comparison principle.