Infinite-dimensional homology and multibump solutions. (English) Zbl 1189.58004

This interesting paper is organized as follows. In Section 2, the authors introduce the Čech homology for compact metric pairs, with coefficients in a field, in a different way from the usual one. They define a homology with compact support for all metric pairs \((X,A)\) as the direct limits of \(\check{H}_*(P,Q)\) with respect to all compact \((P,Q)\subset(X,A)\). These theories satisfy all the Eilenberg-Steenrod axioms. In Section 3, the authors introduce an infinite-dimensional homology and define the notion of critical group in terms of this homology. In Section 4, the Schrödinger equation \(-\Delta u +V(u)u=f(x,u)\), \(u\in H^1(\mathbb R^N)\), where \(V\) and \(f\) are periodic in \(x_1,\dots,x_N\) and 0 is a gap of the spectrum of \(-\Delta +V\) in \(L^2(\mathbb R^N)\), is considered. The authors sketch the procedure of A. Szulkin and T. Weth [J. Funct. Anal. 257, No. 12, 3802–3822 (2009; Zbl 1178.35352)] in order to obtain a ground state for \(f\) of subcritical growth. Section 5 is devoted to the proof that the ground state solution has a nontrivial critical group. Section 6 and Section 7 contain the discussion of multibumb solutions and the related technical details.


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35Q55 NLS equations (nonlinear Schrödinger equations)
55N05 Čech types
58E30 Variational principles in infinite-dimensional spaces


Zbl 1178.35352
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