## 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.

### MSC:

 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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