High-order k-Hessian bifurcation conjecture

Let $\Omega$ be the domain and let $f$ be the nonlinearity in the $k$-Hessian problem

, with asymptotic ratios $f_{0}=0$ and $f_{\infty}=0$. A convex solution means a solution of this problem whose solution function is convex. **High-order k-Hessian bifurcation conjecture.** There \exists a threshold $k^{*}$ such that, whenever $k\ge k^{*}$, there is a positive constant $\lambda_{*}$ for which

has at least two convex solutions for every $\lambda>\lambda_{*}$, exactly one convex solution for $\lambda=\lambda_{*}$, and no nontrivial solutions for every $\lambda\in(0,\lambda_{*})$. This conjecture extrapolates the established multiplicity and threshold behavior in the Monge–Ampère case to sufficiently high $k$ in the $k$-Hessian setting.