Branch-shape and uniqueness conjecture for the semilinear Dirichlet problem in case VII

Let $\Omega$ be the domain and consider the semilinear Dirichlet problem

$$\begin{cases} -\Delta u = u-\lambda u^n-u^2 & \text{in }\Omega,\\ u=0 & \text{on }\partial\Omega, \end{cases}$$

with the parameter regimes and solution branches specified in cases VII2-1 and VII2-2 of Theorem VII1, and with the additional branch supplied by Theorem VIIb.

Branch-shape and uniqueness conjecture. The solution branches in cases VII2-1 and VII2-2 of Theorem VII1 have the same behaviour as the branches depicted in the case-IIb and case-VIIb diagrams, respectively. The branches obtained in Theorem VIIb behave as depicted in the case-VIIc diagram. In case VII2-2, the uniqueness question remains open.

These assertions concern the global shape and multiplicity of branches in parameter regimes where the established results do not determine the complete bifurcation picture. In particular, uniqueness in case VII2-2 is explicitly unresolved.