Abstract
We establish new lower bounds for the number of nodal bound states for the semiclassical nonlinear Schrödinger equation \(-\varepsilon^2 \Delta u+ a(x)u=|u|^{p-2}u\) with bounded and uniformly continuous potential a. The solutions we obtain have precisely two nodal domains, and their positive and negative parts concentrate near the set of minimum points of a. Our approach is independent of penalization techniques and yields, in some cases, the existence of infinitely many nodal solutions for fixed \(\varepsilon\). Via a dynamical systems approach, we exhibit positively invariant sets of sign changing functions for the negative gradient flow of the associated energy functional. We analyze these sets on the cohomology level with the help of Dold’s fixed point transfer. In particular, we estimate their cuplength in terms of the cuplength of equivariant configuration spaces of subsets of \(\mathbb{R}^N\). We also provide new estimates of the cuplength of configuration spaces.
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Albrecht Dold und Dieter Puppe gewidmet: Diese im Jahr 2004 entstandene Arbeit war ursprünglich dem 50. Doktorjubiläum von Albrecht Dold und Dieter Puppe gewidmet. Beide promovierten im Jahr 1954 an der Universität Heidelberg bei Herbert Seifert. T.B. und M.C. verdanken ihnen sehr viel.
In memoriam Dieter Puppe: Dieter Puppe verstarb am 13.8.2005.
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Bartsch, T., Clapp, M. & Weth, T. Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation. Math. Ann. 338, 147–185 (2007). https://doi.org/10.1007/s00208-006-0071-1
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DOI: https://doi.org/10.1007/s00208-006-0071-1