6 problems

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 $2\pi/3$.

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…

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…

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