Existence and nonexistence of entire solutions to the logistic differential equation

Abstract

We consider the one-dimensional logistic problem ${\left(r^{\alpha }A\left(\left|u^{\prime }\right|\right)u^{\prime }\right)}^{\prime }=r^{\alpha }p\left(r\right)f\left(u\right)$ on $\left(0,\infty \right)$, $u\left(0\right)>0$, $u^{\prime }\left(0\right)=0$, where $\alpha$ is a positive constant and $A$ is a continuous function such that the mapping $tA\left(\left|t\right|\right)$ is increasing on $\left(0,\infty \right)$. The framework includes the case where $f$ and $p$ are continuous and positive on $\left(0,\infty \right)$, $f\left(0\right)=0$, 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.