  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.

Analysis of PDEs