Existence and nonexistence of entire solutions to the logistic differential equation. (6th November 2003) - Ghergu, Marius; Rădulescu, Vicenţiu
Abstract : We consider the one-dimensional logistic problem( r α A ( | u ′ | ) u ′ ) ′ = r α p ( r ) f ( u ) on( 0, ∞ ), u ( 0 ) > 0, u ′ ( 0 ) = 0, whereα is a positive constant andA is a continuous function such that the mappingt A ( | t | ) is increasing on( 0, ∞ ) . The framework includes the case wheref andp are continuous and positive on( 0, ∞ ), f ( 0 ) = 0, andf 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 ofp andA . As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.