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An infinite dimensional Saddle Point Theorem and application.

Fabrice Colin1, Ablanvi Songo2

  • 1School of Engineering and Computer Science, Laurentian University, Sudbury, Ontario Canada.

Boundary Value Problems
|March 20, 2026
PubMed
Summary

Researchers developed a new Saddle Point Theorem for strongly indefinite functionals using the τ-topology. This advances the study of solutions for strongly indefinite semilinear Schrödinger equations.

Keywords:
Generalized Saddle Point TheoremIndefinite functionalsSchrödinger equationsτ-topology

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Area of Science:

  • Mathematical Analysis
  • Nonlinear Functional Analysis
  • Partial Differential Equations

Background:

  • Strongly indefinite functionals are crucial in analyzing nonlinear problems.
  • Existing methods for strongly indefinite functionals have limitations.
  • Semilinear Schrödinger equations often involve such functionals.

Purpose of the Study:

  • To establish a new version of the Saddle Point Theorem for strongly indefinite functionals.
  • To apply this theorem to demonstrate the existence of solutions for a specific class of Schrödinger equations.

Main Methods:

  • Utilizing the τ-topology developed by Kryszewski and Szulkin.
  • Developing a novel Saddle Point Theorem tailored for strongly indefinite functionals.
  • Applying the abstract theorem to a concrete semilinear Schrödinger equation.

Main Results:

  • A new, natural version of the Saddle Point Theorem for strongly indefinite functionals has been established.
  • The existence of solutions is proven for a strongly indefinite semilinear Schrödinger equation.
  • The functional J(u) = 1/2⟨Lu, u⟩ - Ψ(u) on a Hilbert space X, with L being a self-adjoint operator with infinite-dimensional negative and positive eigenspaces, is analyzed.

Conclusions:

  • The developed Saddle Point Theorem provides a powerful tool for analyzing strongly indefinite functionals.
  • This work contributes to the understanding of solutions for nonlinear partial differential equations, specifically Schrödinger equations.
  • The τ-topology offers a promising framework for future research in variational methods.