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

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 linearized eigenvalue problem at an unstable solution, find sufficient conditions ensuring that implicit function theorem is applicable one, and then deduce uniqueness result solution. An application our results logistic condition arising from population genetics given.