High-order k-Hessian solution multiplicity 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 nontrivial solution is a solution that is not identically zero. **High-order k-Hessian solution multiplicity conjecture.** There \exists $k^{*}>1$ such that, whenever $k\ge k^{*}$, there is a constant $\lambda_{*}>0$ for which

has no nontrivial solution for every $\lambda\in(0,\lambda_{*})$ and at least two nontrivial solutions for every $\lambda\in(\lambda_{*},+\infty)$. The conjecture contrasts the threshold behavior expected for sufficiently high-order $k$-Hessian equations with the semilinear example discussed in the surrounding text and the known Monge–Ampère result.