9 problems

Euclidean Strichartz conjecture. There holds

Euclidean Strichartz conjecture. The estimate

Brezis–Merle sup-plus-inf conjecture. Under these assumptions, the displayed estimate should hold uniformly in $n$. This is the singular two-dimensional analogue of sup-plus-inf in…

Let $(u,v)$ be an entire solution of the Keller–Segel system on $\mathbb{R}^2\times\mathbb{R}$, meaning that $u$ and $\nabla v$ are smooth and satisfy the system, with the second e…

Let $B_1\subset\mathbb R^2$ be the unit ball, and consider the Dirichlet problem … with $\lambda>0$ and $\gamma_1\in\mathbb N$. A sequence $u_k$ with $\lambda_k\to0^+$ is a blow-up…

Let $u_1$ and $u_2$ satisfy the hypotheses of the preceding sphere covering theorem, except that radial symmetry is not assumed, and let the disk $B_r$ be replaced by a general smo…

Let $u$ be a solution of the Liouville equation … in $\mathbb{R}^2$. The concave-solution conjecture. If $u$ is concave, then $u$ is one-dimensional. The paper proves the analogous…

Let $u$ be a solution of the Liouville equation … in $\mathbb{R}^2$, and let $delta$ be the standard Euclidean metric. Assume that $\mathbb{R}^2$ has diameter $pi$ under the metric…

Brézis–Merle mass quantisation conjecture. Under some mild regularity assumptions on the functions $V_n$ in Theorem A, all the numbers $alpha_p$ are positive integers.