Let Ω be a bounded domain in ℝ2. The study, begun in Keady , of the boundary-value problem, for (λ/k, ψ),
is continued. Here Δ denotes the Laplacian, H is the Heaviside step function and one of λ or k is a given positive constant. The solutions considered always have ψ > 0 in Ω and λ/k > 0, and have cores
In the special case Ω = B(0, R), a disc, the explicit exact solutions of the branch τe have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains Ω and solutions with connected cores A.
An adaptation of the maximum principles and of the domain folding arguments of Gidas, Ni and Nirenberg  is an important step in establishing the above result.