Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

TL;DR

This paper investigates the uniqueness of positive solutions for the Laplace equation with indefinite superlinear boundary conditions, using spectral analysis and the implicit function theorem, with applications to population genetics.

Contribution

It provides new sufficient conditions for the uniqueness of positive solutions in indefinite superlinear boundary value problems, extending previous results.

Findings

Identifies conditions ensuring the uniqueness of positive solutions.

Applies spectral analysis to the linearized eigenvalue problem.

Demonstrates relevance to logistic models in population genetics.

Abstract

In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions and study the uniqueness of positive solutions that this problem possesses. Superlinear elliptic problems can be expected to have multiple positive solutions under certain situations. To our end, by conducting spectral analysis for the linearized eigenvalue problem at an unstable positive solution, we find sufficient conditions for ensuring that the implicit function theorem is applicable to the unstable positive one. An application of our results to the logistic boundary condition arising from population genetics is given.