S-shape multiplicity conjecture for the case III semilinear elliptic problem

Let $\underline{\lambda}<\overline{\lambda}$ be the endpoints of the parameter interval obtained in Theorem

$$for problem$$

.

S-shape conjecture. The interval $(\underline{\lambda},\overline{\lambda})$ should be expandable so that problem

has at least three solutions for every $\lambda\in(\underline{\lambda},\overline{\lambda})$ and at least two solutions for $\lambda=\underline{\lambda}$ and $\lambda=\overline{\lambda}$. Equivalently, the solution branch should have an $S$-shape. The assumption on the size of $\Omega$ used in Theorem

should be removable.

This conjecture predicts an $S$-shaped bifurcation diagram in case III. The supplied passage gives no resolution of either the multiplicity prediction or the domain-size assumption.