23 problems

Let $H$ act on $L^2((0,\pi),\mathbf C^2)$ by … with boundary conditions $f_1(0)=0$, $f_2'(0)=0$, $\cos(\theta)f_1(\pi)+\sin(\theta)f_2(\pi)=0$, and…

Let $L f(x)=A(x)f'(x)$ on $L^2((\alpha,\beta),\mathbf C^n)$, where $A(x)$ is invertible and diagonalizable for each $x\in(\alpha,\beta)$, with boundary conditions $f(\alpha)=u$ and…

Multiple minimizers via symmetry. The constrained problem admits at least three distinct minimizers related by rotations through $2\pi/3$.

Let $u$ be a two-dimensional minimizing solution of the vectorial Allen–Cahn system. Suppose its blow-down limit consists of exactly two phases $a_1$ and $a_2$, separated by a stra…

Lewin's conjecture. The corresponding classification and uniqueness results should generalize to the system for every $N\geq 3$: normalized solutions should necessarily satisfy…

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.…

Existence conjecture. There exists $1<\underline{m}<(p^*)'$ such that, if $f\in L^m(\Omega)$ with $m\geq\underline{m}$, then system $$ admits a weak solution $(u,\varphi)\in W^{1,p…

Consider diagonal systems in the critical case $p=n=2$, with bounded weak solutions and the usual natural growth and ellipticity assumptions. The smallness condition $K_0M<\lambda$…

Let $u,v>0$ solve the critical Hardy–Littlewood–Sobolev system … or the corresponding fractional PDE system … in $\mathbb{R}^n$, where $0<p,q<\infty$ and…

Let $\Omega\subset\mathbb{R}^N$ and consider the Hénon-Lane-Emden system … where $p,q$ satisfy the subcriticality condition in the source and $\alpha,\beta\geq 0$. A least energy n…

Let … Assume that $p>2$ and that $G(u)=\int_{B_{r}}\gamma(\nabla u)\,dx$, where $\gamma$ satisfies the hypotheses specified in the paper and the $p$-growth bounds … The mapping…

Consider the general $SU(3)$ Toda system studied in the paper, with the parameter values $\mu_n$ defined in the preceding discussion. Lemma 6 describes bifurcation from the radial…

Sharp global estimate conjecture. Estimate should hold for all both-singular solutions not only when $N=4$ with $0<\beta<3\max\{\mu_1,\mu_2\}$, but also when $N=3$ without assuming…

Let $D=\operatorname{Int}\bigcup_{g\in G_a}g\overline{F}$, where $F$ is a fundamental region for the action of a finite reflection group $G$, and let $\nu$ be a direction in the in…

Let $G$ be a finite reflection group, let $W$ be invariant under $G$, and let $u:\mathbb{R}^n\to\mathbb{R}^m$ be a $G$-equivariant solution of the elliptic system $\Delta u=W_u(u)$…

Let $\alpha\in(0,n)$, let $p,q$ be positive exponents, and consider the Hardy–Littlewood–Sobolev system … Define the critical Sobolev hyperbola by…

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…

Consider the elliptic system … Let $N\geq 2$, and let $(u,v)$ be a solution satisfying … with the limits uniform in $x'\in\mathbb{R}^{N-1}$. Berestycki–Lin–Wei–Zhao's Gibbons-type…

Consider problem … left{left((1+2nu)^{-1/2}zmu^1,(1+2nu)^{-1/2}zmu^1right):mu0right} … . The theorem preceding this conjecture proves the analogous assertion for positive ground st…

Consider the elliptic system … in Euclidean space, together with the higher-dimensional constructions and properties discussed for this system and for the Allen–Cahn equation. High…

Hardy–Hénon Liouville conjecture. The system has no positive solution in $\mathbb{R}^N$ if and only if

Strict copositivity conjecture. Strict copositivity of $B$ should be sufficient to ensure that the system admits no nontrivial solution.

Bidaut-Véron–Szulkin conjecture. System $(SH)$ with $a=b=0$ admits no radial or nonradial global solutions if and only if $(\delta,\mu)$ lies under the hyperbola