Abstract

We discuss how spatial degeneracy influences the asymptotic profiles and uniqueness of positive solutions to reaction–diffusion equations. The first part is concerned with an elliptic problem involving local crowding and global competition. The common zero set of the two regulation coefficients determines a spectral threshold. Above this threshold, positive solutions blow up inside the common refuge and, after rescaling, converge to a local logistic profile. The second part treats a local elliptic problem with an indefinite weight and nonmonotone regulation. For a fixed parameter in the admissible spectral interval, either a sufficiently small regulation amplitude or a sufficiently small ball containing the regulation region gives a common lower bound for all positive solutions on that region. This places the solutions in the increasing range of the full regulating factor and yields uniqueness.