Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach

TL;DR

This paper investigates the uniqueness and boundary behavior of positive solutions to a nonlinear boundary blow-up problem for a logistic equation, employing Karamata regular variation theory to analyze the asymptotic properties.

Contribution

It introduces a novel application of Karamata regular variation theory to analyze boundary blow-up solutions of nonlinear elliptic equations, providing new insights into their uniqueness and asymptotic expansion.

Findings

Established conditions for uniqueness of solutions.

Derived asymptotic expansion properties near the boundary.

Connected parameter values to eigenvalue problems.

Abstract

We study the uniqueness and expansion properties of the positive solution of the logistic equation $\Delta u+au=b(x)f(u)$ in a smooth bounded domain $\Omega$, subject to the singular boundary condition $u=+\infty$ on $\partial\Omega$. The absorption term $f$ is a positive function satisfying the Keller--Osserman condition and such that the mapping $f(u)/u$ is increasing on $(0,+\infty)$. We assume that $b$ is non-negative, while the values of the real parameter $a$ are related to an appropriate semilinear eigenvalue problem. Our analysis is based on the Karamata regular variation theory.