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The Hénon equation beyond the critical Sobolev exponent

  • Francesca Gladiali
  • Massimo Grossi
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Abstract

We consider the Dirichlet problem −Δu=|x|αup, u>0 in Ω, u=0 on ∂Ω, where Ω⊂ RN, N≥ 3, is a bounded domain containing the origin and α>0. We prove that a solution exists for every exponent 1<p<pα=(N+2+2α)/(N−2), in particular in the whole Sobolev-supercritical range (N+2)/(N−2)≤ p<pα, provided Ω is invariant under O(N−1)× O(1) and satisfies an angular monotonicity condition. This class contains the balls, the oblate ellipsoids of revolution, lens-shaped domains, and non-convex, non-smooth and non star-shaped domains. The solution is obtained by minimizing the Rayleigh quotient on a suitable closed convex cone.

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Cite

BibTeXRIS
BibTeX
@misc{gladiali2026hénon,
  title = {{The Hénon equation beyond the critical Sobolev exponent}},
  author = {Francesca Gladiali and Massimo Grossi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39976},
  eprint = {2610.39976},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39976},
}
RIS
TY  - UNPB
TI  - The Hénon equation beyond the critical Sobolev exponent
AU  - Francesca Gladiali
AU  - Massimo Grossi
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39976
UR  - https://arxiv.org/abs/2610.39976
AB  - We consider the Dirichlet problem −Δu=|x|^αu^p, u>0 in Ω, u=0 on ∂Ω, where Ω⊂ R^N, N≥ 3, is a bounded domain containing the origin and α>0. We prove that a solution exists for every exponent 1<p<p_α=(N+2+2α)/(N−2), in particular in the whole Sobolev-supercritical range (N+2)/(N−2)≤ p<p_α, provided Ω is invariant under O(N−1)× O(1) and satisfies an angular monotonicity condition. This class contains the balls, the oblate ellipsoids of revolution, lens-shaped domains, and non-convex, non-smooth and non star-shaped domains. The solution is obtained by minimizing the Rayleigh quotient on a suitable closed convex cone.
ER  -