44 problems

Let $(M,g)$ be a compact Riemann surface with volume $1$, let $u_k$ be a sequence of blow-up solutions of the mean field equation with $S=\emptyset$, and let $\mathcal{B}$ be its f…

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

Let $\Omega$ be a convex bounded domain in $\mathbb{R}^3$ with smooth boundary. For a sequence $w_{p_n}$ associated with $p_n\to\infty$, let $S$ denote the blow-up set of a subsequ…

Let $u_0\in L^2$ be initial data for the one-dimensional quintic nonlinear Schrödinger equation, and consider a solution $u$ that develops mass concentration near a blow-up time…

Let $u$ be a finite-time blow-up solution in the near-soliton (single-bubble) mass regime for the mass-critical nonlinear Schrödinger equation, with blow-up time $T$ and scale para…

CFG08 extension conjecture. The proof of the analogous blow-up assertion can be carried out without assuming $u(0,t)=\max_{\Omega}u(\cdot,t)$, provided that the conclusions of Lemm…

Exponential-case blow-up conjecture. The blow-up behavior of solutions is close, in some sense, to the blow-up behavior in the exponential case $f(u)=e^u$. This is motivated by the…

Let $\Omega$ be a general bounded domain, and consider the Brezis–Nirenberg problem. A bubble-tower solution is a solution with multiple bubbles concentrating at the same point. Bu…

Let $u_k\in W^{1,2}(\mathbb{B}^2)$ be positive solutions of … on the Poincaré disk, satisfying the boundary decay and uniform energy bound in the stated setting, with…

Blow-up analysis conjecture. A blow-up analysis can be performed in the quasilinear context to prove the existence of extremal functions for $\Phi$ in ${\mathcal B}_1$. This concer…

Boundary concentration quantization conjecture. The concentration constants are all equal to

Let $\Omega$ be a general planar domain, without any symmetry assumption, and consider the Lane–Emden problem in which $p\to+\infty$. A cluster of bubbles is a solution whose bubbl…

Let $R$ be a function satisfying the periodicity and critical-point hypotheses above, with expansion exponent $\beta$ and homogeneous dimension $Q$. Multiple-blowup conjecture. If…

Existence conjecture. There exist sign-changing type II blow-up solutions in the lower dimensions

For $x^*\in\Omega\setminus\Sigma$, consider the blow-ups $v^{x^*,r}\in S^\infty(Q)$ of $v=\nabla u$ defined by … For $\mathcal{H}^1$-almost every $x^*\in\Omega\setminus\Sigma$, com…

Strong-convergence conjecture. The renormalized sequence converges strongly:

Let $u_k$ be a sequence of blow-up solutions of the singular Liouville equation considered in the paper, with a singular source of strength $\alpha>-1$ and…

Rigidity conjecture. Then $\theta_1(t)\to-i$, $\theta_2(t)\to0$, $\lambda_1(t)\to1$, and

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…

Paraboloid blow-up conjecture. For $C^\infty$ coefficients, paraboloids do occur as coincidence sets of blow-up limits with moving centers approaching singularities.

Consider stochastic nonlinear Schrödinger equations in the mass-critical and mass-supercritical regimes, and stable blow-up solutions with slightly supercritical mass in the mass-c…

Brézis–Merle mass quantisation conjecture. Under some mild regularity assumptions on the functions $V_n$ in Theorem A, all the numbers $alpha_p$ are positive integers.

Let $(u_p)$ be a family of positive solutions of the planar elliptic Neumann problem on a bounded domain $\Omega$, satisfying the energy bound, and let $m$ denote the number of bou…

Constant-bound conjecture. The right-hand sides in the inequalities of Theorem and Corollary should be constant rather than powers of the logarithmic time variable; equivalently, t…

Let $\Omega$ be the planar domain and let $u_\varepsilon$ be a family of sign-changing solutions under consideration, blowing up at $\xi_0$ with non-trivial residual mass $u_0$ as…