43 problems

Sharp-threshold conjecture. The sharp threshold for the existence of collapsing ring solutions is $p<5$ in any dimension $N\geq 2$.

Spherically symmetric scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_{0,\operatorname{rad}}=+\infty$, while in the focusing case ($\mu=-1$) we have…

Critical-norm blow-up conjecture. Finite-time blow-up implies

Let $R>0$, let $k\geq 1$ be an integer, and consider the radial problem at the resonant value $\lambda=\lambda_k^{\mathrm{rad}}$. Let $\mu_k^\star(\lambda)$ denote the threshold pa…

Killip–Oh–Pacovnicu–Visan conjecture. There exists $\mu_0\in(0,\mu_*)$ such that the map $\mu\mapsto c_-(\mu)$ is strictly decreasing on $(0,\mu_0)$ and strictly increasing on…

Let $k\geq 2$, $r_0>0$, and define … where $G(t)=\int_0^t g(s)\,ds$. In Theorem 2, there are positive numbers $\underline{r_0}$, $r_0^{**}$ and $\overline{r_0}$ satisfying the stat…

Let $k\geq 2$ and let $u$ be the unique radial solution in Theorem 1 for initial value $u(0)=\xi$, with positive thresholds $\underline{\xi}$, $\xi^{**}$ and $\overline{\xi}$ satis…

Let $B_R\subset\mathbb{R}^N$ be the ball of radius $R$, let $2^*=\frac{2N}{N-2}$, and let $(u_k)_{k\geq 0}$ be radial solutions of the critical Lin–Ni–Takagi problem with exponent…

Let $\lambda_*$ and $\beta_*$ be limiting parameters for the semilinear elliptic problem, with $\lambda_*>\Lambda_1$ and $\beta_*\leq1$. The alternatives (iii)–(vi) refer to the co…

Let $\Lambda^*<\lambda_*<\Lambda_1$ and set $\beta_*=1$. Let $S_{1,\lambda,\beta}$ denote the class of radial sign-changing solutions with the index specified in the paper, and let…

Consider positive solutions of … under the condition $f_r(r,u)\leq 0$. In dimension $n=1$, the value $u(0)$ is known to be a global parameter. Global-parameter conjecture. The same…

Consider the focusing energy-critical wave equation … in odd space dimension $N\geq 3$, with radial initial data in the energy space…

Radial Neumann solution conjecture. If

Consider the planar dynamical system associated with the radial solutions of the scaling-invariant nonlinear elliptic equation, with dimension parameter $N=2$. A cycle means a peri…

Assume $N\geq 3$, let $M<0$, and consider positive radial solutions $u$ with the asymptotic profile $U_M(r)$ as $r\to0$. A solution has the far-field behavior…

Non-$L^2$ conjecture. The solution $u$ does not belong to $L^2(\mathbb{R}^N)$. This extends the corresponding non-$L^2$ conclusion known for the case $\gamma=0$; the assertion for…

Let $A_k$ be the $k\times k$ tridiagonal matrix whose diagonal, subdiagonal and superdiagonal entries are respectively … with the indices as defined in the source, and set…

Let $\mathcal{B}_i^+$ denote the component of the radial bifurcation branch $\mathcal{B}_i$ along which $u_\mu(0)>1$, for $i\geq 2$. The paper considers radial Keller–Segel solutio…

Let $\mathcal{B}_i^-$ denote the component of the radial bifurcation branch $\mathcal{B}_i$ along which $u_\mu(0)<1$. The cited constructions produce radial solutions concentrating…

Let $B_1\subset\mathbb{R}^n$ be the unit ball, and let $\mathcal{B}_i^-$ and $\mathcal{B}_i^+$ denote the two components of the bifurcation branch $\mathcal{B}_i$ of radial solutio…

Consider the bifurcation problem in the parameter $q$ for $p>2$, where the constant solution undergoes bifurcation at $q=p$. Infinite-branch bifurcation conjecture. Infinitely many…

Let $\chi>0$, $k\in(-N,0)$, and $m=2N/(2N+k)$ in dimension $N\geq1$. Assume that the initial datum $\rho_0\in{\mathcal Y}$ is radially symmetric. Global-existence-versus-blow-up co…

Let $\chi>0$, $k\in(-N,0)$, and $m=2N/(2N+k)$ in dimension $N\geq1$. Assume that the initial datum $\rho_0\in{\mathcal Y}$ is radially symmetric, and let $\rho_{HLS,\lambda,c^*}$ d…

Consider the radial Dirichlet problem … with $\lambda>0$. Two-solution conjecture. There are still exactly two positive solutions for $\lambda$ sufficiently large. This is a numeri…

Consider the radial Dirichlet problem … where $\lambda>0$. A solution curve in the $(\lambda,u(0))$ plane has $\lambda\to\infty$ as $u(0)\to0$ and as $u(0)\to\beta$, with a horizon…