Bifurcation-diagram conjecture for the case VIII semilinear elliptic problem

Assume $\lambda_1\geq 1$ and $m+1=2<n<2^*-1$, and consider problem

with parameter $\lambda$. **Case VIII bifurcation conjecture.** The branches of solutions to

should behave as depicted in Figure

$$. Theorem$$

establishes existence of a positive solution for every $\lambda<0$ and nonexistence for every $\lambda\geq0$. The conjecture concerns the detailed shape of the corresponding solution branch, which is not resolved in the supplied passage.