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Journal of Fixed Point Theory and Applications

, Volume 19, Issue 2, pp 1315–1363 | Cite as

Morse–Floer theory for superquadratic Dirac equations, I: relative Morse indices and compactness

  • Takeshi IsobeEmail author
Article

Abstract

In this paper and its sequel (Isobe in Morse–Floer theory for superquadratic Dirac equations, II: construction and computation of Morse–Floer homology, 2016), we study Morse–Floer theory for superquadratic Dirac functionals associated with a class of nonlinear Dirac equations on compact spin manifolds. We are interested in two topics: (i) relative Morse indices and its relation to compactness issues of critical points; (ii) construction and computation of the Morse–Floer homology and its application to the existence problem for solutions to nonlinear Dirac equations. In this part I, we focus on the topic (i). One of our main results is a compactness of critical points under the boundedness assumption of their relative Morse indices which is an analogue of the results of Bahri–Lions (Commun Pure Appl Math 45:1205–1215, 1992) and Angenent–van der Vorst (Math Z 231: 203–248, 1999) for Dirac functionals. To prove this, we give an appropriate definition of relative Morse indices for bounded solutions to \(\mathsf {D}_{g_{\mathbb {R}^{m}}}\psi =|\psi |^{p-1}\psi \) on \(\mathbb {R}^{m}\). We show that for \(m\ge 3\) and \(1<p<\frac{m+1}{m-1}\), the relative Morse index of any non-trivial bounded solution to that equation is \(+\infty \). We also give some useful properties of the relative Morse indices of Dirac functionals which will be used in the study of the topic (ii) above.

Keywords

Relative Morse index Compactness Dirac equations 

Mathematics Subject Classification

35Q41 37B30 57R58 58E05 

Notes

Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers 22540222, 15K04947.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of Science and EngineeringTokyo Institute of TechnologyTokyoJapan

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