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Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation. (English) Zbl 1126.35061

The authors establish lower bounds for the number of nodal bound states for the semiclassical nonlinear Schrödinger equation \(-\varepsilon ^{2}\Delta u+a\left( x\right) u=\left| u\right| ^{p-2}u\) in \({\mathbb R}^{N},\) with bounded and uniformly continuous potential \(a.\) The solutions have two nodal domains, and their positive and negative parts concentrate near the set of minimum points of \(a.\) The existence of infinitely many nodal solutions for fixed small \(\varepsilon \) is obtained. Variational techniques and Lusternik-Schnirelman theory are used. The paper provides the lower bounds for the cuplength of the configuration spaces.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J20 Variational methods for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
55R80 Discriminantal varieties and configuration spaces in algebraic topology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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