20 problems

Let $\gamma>2$ and consider the first excited states arising from the ground state at the bifurcation point for the logarithmic Schrödinger equation with a $\delta^{\prime}$-intera…

Hessian determinant conjecture. The determinant satisfies

Let $u_{c,\kappa}$ be a black or gray soliton solving the equation … partialcP(u{c,kappa})<0, … partialcP(u{c,kappa})0. … , whose skew-symmetric operator is not surjective. The ins…

Let $B_1$ be the unit ball, let $\beta$ denote the first Dirichlet eigenvalue parameter, and let $Q_b$ be the least-energy (ground state) solution of the stationary nonlinear Schrö…

Stability conjecture. Let $1\leq d\leq 6$. For $\alpha\leq 2/d$, the solution $\varphi_\omega$ minimizes $\mathcal{E}$ at mass $M=M(\omega)$ for all $\omega\in(0,\omega^*)$ and is…

Let $m_\mu$ denote the mountain pass level, and let the standing wave corresponding to a solution at level $m_\mu$ be considered for $0<\mu<\mu_1$. Perturbative instability conject…

Let $V$ be the attractive inverse power potential considered in the paper, and let $\phi_\omega$ be the unique positive least action ground state. Here $\omega_0$ is the lower endp…

For $d=2$, let $Q_\omega$ denote a cubic-quintic ground state solution at frequency $\omega\in(0,\tfrac{3}{16})$. Orbital stability conjecture. All cubic-quintic ground state solut…

Stability-transition conjecture. The ground state $u_{\widetilde{T}}$ is stable up to a critical value of $\omega$ and becomes unstable thereafter.

Uniqueness conjecture. Uniqueness holds for energy minimizers associated with more general nonlinearities; equivalently, for each fixed mass $\rho$, the constrained minimization pr…

In dimension $d=3$, let $\phi_\omega$ denote the ground state solution of the cubic-quintic Schrödinger equation at frequency $\omega$, with $0<\omega<\tfrac{3}{16}$, and let…

Consider the one-dimensional nonlinear Schrödinger equation with attractive delta potential, with exponent $p>5$, and let $Q_\omega$ denote the solitary-wave profile. Let…

Let $\lambda_0$ and $\lambda_1$ be the lower bounds appearing in Theorems 1 and 2 for the parameter $\mu$, and let a ground state of equation (1.3) with $\mu=0$ determine these bou…

Let $m_1$ denote the parameter defining the masses in the 2DF setting, and let Broucke's orbit be the periodic orbit under consideration. Conjecture on Broucke's orbit. Broucke's o…

Excited-state instability conjecture. Every excited state for $N\geq3$ and $\omega\in\mathbb{R}$ is unstable in $W(\Gamma)$.

Let a standing wave be a standing-wave solution associated with a critical point of the constrained functional, and distinguish mountain pass solutions from local minimizers. Orbit…

Invariant Gaussian-factor conjecture. There exists a unique solution of the linearized log-KdV equation of the form

Uniqueness conjecture. The minimizer $U(x)$ of the energy $E$ in $X$ is unique, up to translation and gauge invariance.

Let $N$ be the spatial dimension, let $1<p<3+\frac{4}{N}$, and consider the quasi-linear Schrödinger equation … A ground state solution is a solution minimizing the associated grou…

Let $(a,E,c)\in\Omega$ parameterize a periodic traveling wave of the generalized Korteweg–de Vries equation with a power-law nonlinearity. Write $T_E$ for the derivative of the per…