6 problems

Fully nonlinear Yamabe conjecture. There exists a positive smooth function $u\in C^\infty(M^n)$ such that

Concentration conjecture. As $\gamma$ varies in the stated regime, $u_\gamma$ converges simply to the characteristic function of the point $\{0\}$.

Consider the power-like Hamiltonian $H=A|Du|^\gamma$ with $A<0$ and $\gamma\neq2$. Negative-coefficient Liouville conjecture. The Liouville property should hold as in the quadratic…

Let $u$ be a nonnegative solution of … in $\mathbb{R}^N$, where $\beta$ is the parameter associated with the Pucci operator. Liouville conjecture. Any such solution is constant pro…

Blueprint conjecture. The strategies developed for the previously studied free transmission models should provide a solid blueprint for studying free transmission problems governed…

Let $u:{\mathbb R}\times{\mathbb C}^n\to{\mathbb R}$ be real-valued, and set … where $\partial\bar\partial u$ is an $n\times n$ Hermitian matrix. For $1\leq k\leq n$, define … with…