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

TL;DR

This paper proves the existence and uniqueness of positive solutions for a class of 1-D diffusive logistic boundary value problems with general boundary conditions, and analyzes their behavior as parameters vary.

Contribution

It establishes the existence and uniqueness of positive solutions for a broad class of boundary conditions, including non-classical and non-monotonic cases, and characterizes their asymptotic behavior.

Findings

Existence of positive solutions under general boundary conditions.

Uniqueness of positive solutions for constant coefficient cases.

Asymptotic behavior of solutions as parameters tend to infinity.

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 $-u'(0)+\beta u(0)=0$.