Existence and nonexistence of entire solutions to the logistic differential equation (Q1773481)

The authors study the following problem \[ [r^{\alpha}A(| u'| )u']'=r^{\alpha}p(r)f(u), \quad r>0, \quad u(0)>0, \quad u'(0)=0, \] where \(\alpha>0\) and \(A\) is a continuous function such that the map \(t\mapsto tA(| t| )\) 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. First, it is established a general nonexistence result on this problem. Also several existence or nonexistence results for large solutions are given. As a consequence, it is deduced that the mean curvature inequality problem in \(\mathbb{R}^n\) does not have nonnegative solutions, except the trivial one.