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<p>By using the <i>τ</i>-topology of Kryszewski and Szulkin, we establish a natural new version of the Saddle Point Theorem for strongly indefinite functionals. The abstract result will be applied to study the existence of solutions to a strongly indefinite semilinear Schrödinger equation, where the associated functional is indefinite, that is, the functional is of the form <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>J</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mo stretchy="false">〈</mo> <mi>L</mi> <mi>u</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">〉</mo> <mo>−</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$J(u) = \dfrac{1}{2} \langle Lu, u \rangle - \Psi (u)$</EquationSource> </InlineEquation> defined on a Hilbert space <i>X</i>, where <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>L</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </math></EquationSource> <EquationSource Format="TEX">$L : X \to X$</EquationSource> </InlineEquation> is a self-adjoint operator whose negative and positive eigenspaces are both infinite-dimensional.</p>

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An infinite dimensional Saddle Point Theorem and application

  • 期刊论文

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  • 发表日期：

    2026年2月11日

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  • Fabrice Colin,
  • Ablanvi Songo

摘要

By using the τ-topology of Kryszewski and Szulkin, we establish a natural new version of the Saddle Point Theorem for strongly indefinite functionals. The abstract result will be applied to study the existence of solutions to a strongly indefinite semilinear Schrödinger equation, where the associated functional is indefinite, that is, the functional is of the form J ( u ) = 1 2 〈 L u , u 〉 − Ψ ( u ) $J(u) = \dfrac{1}{2} \langle Lu, u \rangle - \Psi (u)$ defined on a Hilbert space X, where L : X → X $L : X \to X$ is a self-adjoint operator whose negative and positive eigenspaces are both infinite-dimensional.