In this paper, we further investigate the bifurcation from infinity of the nonlinear SchrÖdinger equation <Equation ID="Equ45"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2340_Article_Equ45.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="291" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} -\Delta u+V(x)u=\lambda u+f(x,u),\hspace{0.4cm}x\in \mathbb {R}^N. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="11.38092pt" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Based on the invariant manifold given by Li and Wang (Nonlinear Anal 213:112490. 2021), we first establish the existence of connecting trajectory between two invariant sets constituting a Morse decomposition for the corresponding parabolic equations on unbounded domains under some assumptions on V and f, which partially extends the results in Kokocki (J Differ Equ 255:1554-1575, 2013), Rybakowski (J Differ Equ 51: 182-212, 1984). Furthermore, we use this dynamical result and the homology and cohomology Conley indices and the Poincaré-Lefschetz duality theory of the Conley index to establish some new multiplicity results of solutions of the system on bifurcations from infinity, which significantly improve the earlier works in the literature and the results obtained in Li and Wang (Nonlinear Anal 213:112490. 2021).

Bifurcation from Infinity of the Schrödinger Equation via Invariant Manifolds II: Higher Multiplicity

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发表日期：

2025年1月4日

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Chunqiu Li,
Jintao Wang

摘要

In this paper, we further investigate the bifurcation from infinity of the nonlinear SchrÖdinger equation $\begin{aligned} -\Delta u+V(x)u=\lambda u+f(x,u),\hspace{0.4cm}x\in \mathbb {R}^N. \end{aligned}$ $\begin{matrix} - Δ u + V (x) u = λ u + f (x, u), x \in R^{N} . \end{matrix}$ Based on the invariant manifold given by Li and Wang (Nonlinear Anal 213:112490. 2021), we first establish the existence of connecting trajectory between two invariant sets constituting a Morse decomposition for the corresponding parabolic equations on unbounded domains under some assumptions on V and f, which partially extends the results in Kokocki (J Differ Equ 255:1554-1575, 2013), Rybakowski (J Differ Equ 51: 182-212, 1984). Furthermore, we use this dynamical result and the homology and cohomology Conley indices and the Poincaré-Lefschetz duality theory of the Conley index to establish some new multiplicity results of solutions of the system on bifurcations from infinity, which significantly improve the earlier works in the literature and the results obtained in Li and Wang (Nonlinear Anal 213:112490. 2021).