We consider the one-dimensional logistic problem
$(r^{\alpha}A(|u'|)u')'= r^{\alpha}p(r)f(u)$ on $(0,\infty)$ ,
$u(0)>0$ , $u'(0)=0$ , where $\alpha$ is a positive constant and
$A$ is a continuous function such that the mapping $tA(|t|)$ is
increasing on $(0,\infty)$ . The framework includes the case where
$f$ and $p$ are continuous and positive on $(0,\infty)$ , $f(0)=0$ , 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.