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Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities

  • Bao Yu
  • Yang Zhou
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Abstract

Let N=m+1 and consider the thin ellipsoid Ωε={(y,xN)∈ℝm×ℝ: |y|2+ε−2xN2<1}. We prove that in every sufficiently large dimension, for every sufficiently small ε>0, there exists a smooth positive, strictly increasing, strictly convex, superlinear nonlinearity fε for which the extremal solution in Ωε is an unbounded H10 solution. Combining this result with Dancer's thin-domain regularity theorem for the Gelfand nonlinearity f(t)=et, we obtain on the same sufficiently thin ellipsoids a bounded Gelfand extremal solution and an unbounded extremal solution for another nonlinearity. Thus, in this two-part sense, Brézis' Open Problem~6.1 is resolved in every sufficiently large dimension.

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Cite

BibTeXRIS
BibTeX
@misc{yu2026singular,
  title = {{Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities}},
  author = {Bao Yu and Yang Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06673},
  eprint = {2609.06673},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06673},
}
RIS
TY  - UNPB
TI  - Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities
AU  - Bao Yu
AU  - Yang Zhou
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06673
UR  - https://arxiv.org/abs/2609.06673
AB  - Let N=m+1 and consider the thin ellipsoid Ω_ε={(y,x_N)∈ℝ^m×ℝ: |y|^2+ε^−2x_N^2<1}. We prove that in every sufficiently large dimension, for every sufficiently small ε>0, there exists a smooth positive, strictly increasing, strictly convex, superlinear nonlinearity f_ε for which the extremal solution in Ω_ε is an unbounded H^1_0 solution. Combining this result with Dancer's thin-domain regularity theorem for the Gelfand nonlinearity f(t)=e^t, we obtain on the same sufficiently thin ellipsoids a bounded Gelfand extremal solution and an unbounded extremal solution for another nonlinearity. Thus, in this two-part sense, Brézis' Open Problem~6.1 is resolved in every sufficiently large dimension.
ER  -