18 problems

Consider semilinear elliptic equations … with $u\in H^1(\mathbb{R}^n)$ and $u\not\equiv0$, under the natural structural conditions on $f$ considered by Berestycki and Lions. A radi…

Lions's conjecture. Neither convexity of the domain $\Omega$ nor the technical condition

Let $a(x)=a_{\kappa,\varepsilon}(x)$ with $\kappa\geq1$ and $\varepsilon<1$ arbitrarily close to $1$. Arbitrarily large multiplicity conjecture. By taking $\kappa$ sufficiently lar…

Persistence conjecture. The inequality $2\leq n\leq6$ can be replaced by

Uniqueness conjecture. Uniqueness holds across the entire range

Zero-set conjecture for non-odd nonlinearities. If the nonlinear function $f$ is not odd, then $a_F$ has a discrete and countable number of zeroes with respect to $x_2$, with

The paper considers normalized solutions of Schrödinger equations on large bounded domains … is not essential for the results. This suggests that the existence and qualitative conc…

Let $e(\rho)$ be the constrained energy minimization problem considered in the paper, and let $\rho^*:=\|Q\|_2^2$, where $Q=Q(|x|)$ is the unique positive solution of the associate…

Classification conjecture. Up to a multiplicative constant, every minimizer of $S$ is of the form $U((x-x_0)/\varepsilon)$ for some $x_0\in\mathbb{R}^N$ and $\varepsilon>0$.

Let $X$ be the function space used for the Poincaré operator, let $\tau$, $\varepsilon$, $x_0$, $\varphi^*$, $\mu$, $\omega$, and $\overline D$ be the parameters and functions defi…

Let $p>2$ and consider the elliptic system studied in the paper, with a fully synchronized solution meaning one whose components share the same profile up to constant coefficients.…

Let $K_4$ be the constant appearing in Theorem 3, and let $σ$ be the parameter used there. Threshold constant conjecture. In Theorem 3, $K_4$ is larger than … The quantity on the r…

Let $\lambda$ be the parameter in the indefinite parameter-dependent Neumann problem, and let $\mu^{*}_0(\lambda)$, $\mu^{*}_1(\lambda)$, $\mu^{**}_0(\lambda)$, and…

Let $\mu$ be a symmetric unimodal distribution of natural frequencies, and let $V_{1}^{\mu}(r_{1},r_{2})$ and $V_{2}^{\mu}(r_{1},r_{2})$ denote the two-community self-consistency m…

Existence conjecture. System $$ has at least one positive non-synchronized solution; in fact, there may be infinitely many such solutions, for arbitrary $beta<0$.

Let $\Omega$ be the domain, let $p$ be the exponent, let $m\geq 1$, and let ${\bf v}(x,t)$ be the solution under consideration. Let $\Upsilon_p$ denote the exponential rescaling ra…

Finiteness conjecture. The upper charge bound satisfies

For each $N>0$, let $J_N(a)$ be the function defined in the paper. Sign-change conjecture. There exists a function $c:(0,+\infty)\to(0,+\infty)$ such that … and … Equivalently,…