Let N≥ 3, let R^N_+=x∈R^N:x_N>0, and let 0∈partialR^N_+. Consider positive solutions of the critical problem [ -Delta u=u^fracN+2N-2quadin R^N_+,qquad u=0 quadon partialR^N_+setminus0, ] with an isolated singularity at 0. The problem asks: Whether there exist positive Delaunay-type log-periodic solutions; in particular, solutions for which some T>0 satisfies [ u(e^T x)=e^-fracN-22Tu(x)quadfor all x∈R^N_+. ] Whether every positive solution with an isolated boundary singularity has the anticipated stationary asymptotic classification near 0, including whether there is a universal constant C such that [ u(x)≤ C,x_N|x|^-N/2 ] for all sufficiently small x∈R^N_+.

Delaunay-solution and singular-classification problems for the critical boundary equation

Let $N\ge 3$ , let $\mathbb{R}^N_+=\{x\in\mathbb{R}^N:x_N>0\}$ , and let $0\in\partial\mathbb{R}^N_+$ . Consider positive solutions of the critical problem

-\Delta u=u^{\frac{N+2}{N-2}}\quad\text{in }\mathbb{R}^N_+,\qquad u=0\quad\text{on }\partial\mathbb{R}^N_+\setminus\{0\},

with an isolated singularity at $0$ . The problem asks:

Whether there exist positive Delaunay-type log-periodic solutions; in particular, solutions for which some $T>0$ satisfies

u(e^T x)=e^{-\frac{N-2}{2}T}u(x)\quad\text{for all }x\in\mathbb{R}^N_+.

Whether every positive solution with an isolated boundary singularity has the anticipated stationary asymptotic classification near $0$ , including whether there is a universal constant $C$ such that

u(x)\le C\,x_N|x|^{-N/2}

for all sufficiently small $x\in\mathbb{R}^N_+$ .

Sources & referencesView supporting material

Primary source

arXiv

Additional references

The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II — arXiv — Hua-Yang Wang, Jingang Xiong

elliptic-equations pde

2 days ago

Progress summary

Refreshed 2 days ago

Claimed solved

A new preprint claims to produce the missing boundary-singularity solutions and to disprove the expected universal classification, but the result has not yet been independently verified.

The problem concerns positive solutions of the critical semilinear equation in a half-space with an isolated boundary singularity. It asks whether periodic concentrating solutions exist and whether all singular solutions have the anticipated stationary asymptotics; these questions were posed by del Pino, Musso, and Pacard, and by Bidaut-Véron, Ponce, and Véron, in 2007.

Known results

Bidaut-Véron, Ponce, and Véron (2007) classified singularities for $q_{BT}(N)<q<q_S(N-1)$ under a scale-invariant estimate, leaving the critical case open.
Caffarelli, Gidas, and Spruck (1989) established the contrasting interior classification theory.

August 2026 preprint

Hua-Yang Wang and Jingang Xiong claim a global branch of positive log-periodic solutions, a uniquely determined blow-up period, and local uniqueness of the concentrating family. They also claim that the stationary asymptotic classification and the bound $u(x)\le Cx_N|x|^{-N/2}$ fail; these claims remain unverified.

Current status (as of August 2026): The preprint claims to settle existence and local uniqueness while refuting the proposed classification and uniform bound, but independent verification is absent.

Sources

arxiv.org

No solutions have been posted yet.