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

Abstract

In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions. Superlinear elliptic problems can be expected to have multiple positive solutions by some case. 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 one, and then deduce the uniqueness result for a positive solution. An application of our results to the logistic boundary condition arising from population genetics is given.