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A Brezis--Merle inequality for the biharmonic operator with Dirichlet boundary conditions in four-dimensional balls

  • João Marcos do Ó
  • Federica Sani
  • Victor Rafael Silva
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Abstract

In this paper, we establish a Brezis-Merle type inequality for very weak solutions of the biharmonic Dirichlet problem in the four-dimensional unit ball. By combining this inequality with moving-plane arguments, we obtain uniform a priori estimates for positive solutions of semilinear biharmonic equations with exponential nonlinearities in both the subcritical and critical regimes. As an application of these estimates, we prove the existence of a positive solution by means of topological methods. We also extend the analysis to a class of non-variational cooperative systems.

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Cite

BibTeXRIS
BibTeX
@misc{do2026brezismerle,
  title = {{A Brezis--Merle inequality for the biharmonic operator with Dirichlet boundary conditions in four-dimensional balls}},
  author = {João Marcos do Ó and Federica Sani and Victor Rafael Silva},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27212},
  eprint = {2610.27212},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27212},
}
RIS
TY  - UNPB
TI  - A Brezis--Merle inequality for the biharmonic operator with Dirichlet boundary conditions in four-dimensional balls
AU  - João Marcos do Ó
AU  - Federica Sani
AU  - Victor Rafael Silva
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27212
UR  - https://arxiv.org/abs/2610.27212
AB  - In this paper, we establish a Brezis-Merle type inequality for very weak solutions of the biharmonic Dirichlet problem in the four-dimensional unit ball. By combining this inequality with moving-plane arguments, we obtain uniform a priori estimates for positive solutions of semilinear biharmonic equations with exponential nonlinearities in both the subcritical and critical regimes. As an application of these estimates, we prove the existence of a positive solution by means of topological methods. We also extend the analysis to a class of non-variational cooperative systems.
ER  -