5 problems

Let $N=2$ and let $m>1$. For each positive mass $\mu>0$, consider the polyharmonic Dirichlet and Navier boundary-value problems under assumptions (FP1)--(FP4), with the normalizati…

Let $B$ be the unit ball of $\mathbb{R}^n$, and let $k\in\mathbb{N}$ satisfy $n>2k\geq 2$. Consider $\lambda\in\mathbb{R}$ and $u\in C^{2k}(\overline{B})$ solving … where…

The polyharmonic heat equation is obtained by replacing the biharmonic operator with a higher-order polyharmonic operator, and its linear counterpart is the corresponding homogeneo…

For $s\in\mathbb{N}$, let $m\in\mathbb{N}$ and define the boundary Poisson kernels … for $k\in\{0,\ldots,m\}$, $y\in\partial\mathbb{R}^N_+$, and $x\in\mathbb{R}^N\setminus\{y\}$. U…

Hé non-Lane-Emden conjecture. If $(u,v)$ is a nonnegative solution of the system and $(p,q)$ is under the critical hyperbola, then $u=v=0$. This is a Liouville-type conjecture for…