  We study the uniqueness of reaction-diffusion steady states in general
domains with Dirichlet boundary data. Here we consider "positive" (monostable)
reactions. We describe geometric conditions on the domain that ensure
uniqueness and we provide complementary examples of nonuniqueness. Along the
way, we formulate a number of open problems and conjectures. To derive our
results, we develop a general framework, the stable-compact method, to study
qualitative properties of nonlinear elliptic equations.
