The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions

This paper proves the uniqueness of the positive linearly stable steady-state for a paradigmatic class of superlinear indefinite parabolic problems arising in population dynamics, under nonhomogeneous Dirichlet conditions on the boundary of the domain. The result is absolutely non-trivial, since examples are known for which the model admits an arbitrarily large number of steady-states. Our proof is based on some local and global continuation techniques. Optimal existence and multiplicity results are also obtained through some additional monotonicity and topological techniques.