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A New Minimax Theorem for a Class of C1-Functionals

  • Denilson S. Pereira
  • João R. Santos Júnior
  • João F. F. Silva
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Abstract

We establish a new abstract minimax theorem for obtaining ground-state critical points of a class of C1-functionals. The central feature of this class is that the associated Nehari set is not necessarily a differentiable manifold, but is homeomorphic to a (possibly non-complete) submanifold of the unit sphere. This occurs for functionals whose fibering maps t ↦ I(tu) possess a unique global maximum point only when u belongs to an open cone in the underlying Banach space. A lower semicontinuous boundary barrier, which may be finite, controls minimizing sequences near the boundary of the spherical cone. As a consequence, we provide a general minimax characterization of the ground-state energy level. Elliptic applications complement existing results and include superlinear and quasilinear problems with finite barriers, including thresholds depending on the profile of a first eigenfunction and its gradient.

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Cite

BibTeXRIS
BibTeX
@misc{pereira2026minimax,
  title = {{A New Minimax Theorem for a Class of $C^1$-Functionals}},
  author = {Denilson S. Pereira and João R. Santos Júnior and João F. F. Silva},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31262},
  eprint = {2610.31262},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31262},
}
RIS
TY  - UNPB
TI  - A New Minimax Theorem for a Class of C^1-Functionals
AU  - Denilson S. Pereira
AU  - João R. Santos Júnior
AU  - João F. F. Silva
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31262
UR  - https://arxiv.org/abs/2610.31262
AB  - We establish a new abstract minimax theorem for obtaining ground-state critical points of a class of C^1-functionals. The central feature of this class is that the associated Nehari set is not necessarily a differentiable manifold, but is homeomorphic to a (possibly non-complete) submanifold of the unit sphere. This occurs for functionals whose fibering maps t ↦ I(tu) possess a unique global maximum point only when u belongs to an open cone in the underlying Banach space. A lower semicontinuous boundary barrier, which may be finite, controls minimizing sequences near the boundary of the spherical cone. As a consequence, we provide a general minimax characterization of the ground-state energy level. Elliptic applications complement existing results and include superlinear and quasilinear problems with finite barriers, including thresholds depending on the profile of a first eigenfunction and its gradient.
ER  -