5 problems

For every $n\ge 2$, $0<s<1$, and $p,q>0$, there do not exist strictly positive functions $u,v:\mathbb{R}^n\to(0,\infty)$ satisfying $(-\Delta)^s u=v^p$ and $(-\Delta)^s v=u^q$ in…

Let $\alpha\in(0,2)$, and let $\{\varphi_n^{(\alpha)}\}_{n\geq 1}$ be an $L^2((-1,1))$-normalized sequence of eigenfunctions of the fractional Laplacian on $(-1,1)$ with the approp…

Let $n$ be the boundary dimension, let $\gamma\in(0,\frac{1}{2}]$ with $n>2\gamma$, and let $(\overline{X},\bar{g})$ be a smooth compactification that is not conformally diffeomorp…

Spectral sign-transform inequality conjecture. For $s\in(1,\frac32)$, the analogous inequalities should hold for spectral quadratic forms. The corresponding inequality has been pro…

Fractional Lieb-Thirring coupling conjecture. The optimal constant also satisfies