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

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.