Large positive solutions for a class of 1-D diffusive logistic problems with general boundary conditions

Abstract

The first goal of this paper is to establish the existence of a positive solution for the singular boundary value problem (1.1), where $\mathcal{B}$ is a general boundary operator of Dirichlet, Neumann or Robin type, either classical or non-classical; in the sense that, as soon as $\mathcal{B}u(0)=-u'(0)+\beta u(0)$, the coefficient $\beta$ can take any real value, not necessarily $\beta\geq 0$ as in the classical Sturm--Liouville theory. Since the function $f(u):=au^p -\lambda u$, $u\geq 0$, is not increasing if $\lambda>0$, the uniqueness of the positive solution of (1.1) is far from obvious, in general, even for the simplest case when $a(x)$ is a positive constant. The second goal of this paper is to establish the uniqueness of the positive solution of (1.1) in that case. At a later stage, denoting by $L_\lambda$ the unique positive solution of (1.1) when $a(x)$ is a positive constant, we will characterize the point-wise behavior of $L_\lambda$ as $\lambda\to \pm \infty$. It turns out that any positive solution of (1.1) mimics the behavior of $L_\lambda$ as $\lambda \to \pm\infty$. Finally, we will establish the uniqueness of the positive solution of (1.1) when $a(x)$ is non-increasing in $[0,R]$, $\lambda\geq 0$, and $\beta<0$ if $-u'(0)+\beta u(0)=0$.