Nonradial entire solutions of the critical Hénon equation for every α>2
Abstract
We consider the critical Hénon equation −Δu=|x|αu(N+2+2α)/(N−2) in ℝN, N≥3. Gladiali, Grossi, and Neves (\emph{Advances in Mathematics}, 2013) constructed nonradial continua bifurcating at αk=2(k−1), k≥2, and conjectured that these continua are confined to the even parameters. Jiang and Xiong (2026) disproved this conjecture by obtaining nonradial solutions in punctured neighborhoods of αk for even k>(N−2)/2. We prove that positive nonradial finite-energy entire solutions exist for every α>2 in every dimension N≥3. The proof introduces a global variational construction by direct minimization on a monotonicity cone on the Emden--Fowler cylinder. Angular and axial order restore compactness in every finite Lr, r>2, despite Sobolev supercriticality, and invariance of the linear resolvent yields the full equation. This constructs a solution at each prescribed parameter without continuing local bifurcation branches. The solutions have Newtonian decay, are invariant under O(N−1)× O(1) and the Kelvin transform, and have energy strictly below the radial level. We also obtain a second solution for every α>6 in even dimensions, nonvertical branches at every αk with k≥2 even without dimension restrictions, a cubic energy law, and rigidity near α=0 under bounded cylindrical amplitude.
Related
- Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities
- Smooth nonradial stationary solutions to SQG via the half-Yamabe equation
- Structure of radial solutions to a Hénon type equation with exponential nonlinearity on the hyperbolic space
- Sufficient conditions for multiplicity and uniqueness of positive solutions of the Hénon equation
- Existence of positive Solutions to Hénon equation on integer Lattice graph and Half-Space
BibTeX
@misc{le2026nonradial,
title = {{Nonradial entire solutions of the critical Hénon equation for every $α>2$}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02420},
eprint = {2611.02420},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02420},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonradial entire solutions of the critical Hénon equation for every α>2
AU - Phuong Le
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02420
UR - https://arxiv.org/abs/2611.02420
AB - We consider the critical Hénon equation −Δu=|x|^αu^(N+2+2α)/(N−2) in ℝ^N, N≥3. Gladiali, Grossi, and Neves (\emph{Advances in Mathematics}, 2013) constructed nonradial continua bifurcating at α_k=2(k−1), k≥2, and conjectured that these continua are confined to the even parameters. Jiang and Xiong (2026) disproved this conjecture by obtaining nonradial solutions in punctured neighborhoods of α_k for even k>(N−2)/2. We prove that positive nonradial finite-energy entire solutions exist for every α>2 in every dimension N≥3. The proof introduces a global variational construction by direct minimization on a monotonicity cone on the Emden--Fowler cylinder. Angular and axial order restore compactness in every finite L^r, r>2, despite Sobolev supercriticality, and invariance of the linear resolvent yields the full equation. This constructs a solution at each prescribed parameter without continuing local bifurcation branches. The solutions have Newtonian decay, are invariant under O(N−1)× O(1) and the Kelvin transform, and have energy strictly below the radial level. We also obtain a second solution for every α>6 in even dimensions, nonvertical branches at every α_k with k≥2 even without dimension restrictions, a cubic energy law, and rigidity near α=0 under bounded cylindrical amplitude.
ER -