## 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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