Multiplicity and branch-shape conjecture for the semilinear Dirichlet problem in case VI

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

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

with $m=1$, $0<n<1$, and first Dirichlet eigenvalue $\lambda_1<1$. Let $\lambda^*>0$ be the threshold from the existence theorem.

Multiplicity and branch-shape conjecture. The branches of solutions behave as depicted in the case-VI bifurcation diagram: the problem has at least two solutions for every $\lambda\in(0,\lambda^*)$ and at least one solution for $\lambda=\lambda^*$. Moreover, it remains open whether the solutions with $\lambda>0$ obtained in the cited result are positive or of compact-support type.

The preceding theorem establishes existence below $\lambda^*$ and nonexistence above it, but does not establish the asserted multiplicity or the precise branch structure. The positivity and compact-support properties of the solutions also remain unresolved.