13 problems

Existence conjecture. The equation

Let $P$ be the polynomial defining the nonlinear eigenvalue problem, let $m\geq 2$ be its degree parameter, and let $c_{j,s}$ be the coefficients in the semiclassical trace expansi…

Let $m$ be a selection function and consider the nonlinear eigenproblem for eigenpairs $(\operatorname{\boldsymbol{\lambda}},\operatorname{\boldsymbol{F}})$ described in the paper.…

Coefficient conjecture. The overall coefficient in this asymptotic formula is $2^{5/6}$.

Let the nonlinear eigenvalue problem referred to as equation be the problem governing the excited states of the Bose–Einstein condensate. The conjecture concerns eigenfunctions of…

Let $a$ be a kernel that decays sufficiently fast at infinity, and define … Let $\xi_\sigma$ be the rescaled support parameter, $\overline{w}_\sigma$ the rescaled profile,…

Let $a$ be a sufficiently regular kernel, and define … Let $\xi_\sigma$ be the rescaled support parameter, $\overline{w}_\sigma$ the rescaled profile, $\overline{\chi}$ the indicat…

Let $L\colon\mathbb{R}^k\to\mathbb{R}^k$ be a linear operator, let $\mathbf{S}$ be the unit sphere in $\mathbb{R}^k$, and let $N\colon\mathbf{S}\to\mathbb{R}^k$ be continuous. For…

Let $b$ be a kernel satisfying Assumption MainKernel, let the nonlinearity satisfy Assumption MainNonl, and let $\mathcal{C}_K$ be the admissible class on which the functional…

Non-polynomial eigenfunction conjecture. No eigenfunction of equation (Orig-Eq) can be a polynomial on any open set in $\mathbb{R}^3$.

Nehari manifold obstruction conjecture. Solutions of $$ cannot be obtained by the Nehari manifold method if

Basic conjecture. The set of extreme values of the Nehari manifold method is contained in

Let $\sigma$ lie in the $L^2$-supercritical and $H^2$-subcritical regime, namely … Consider the radial nonlinear eigenvalue problem for $\tilde S_{\mathrm B}(\rho)$ with $\nu=1$: ……