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Localized nodal solutions for semiclassical nonlinear Schroedinger equations Wang, Zhi-Qiang


We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schr\"odinger equation $−\epsilon^2 \Delta v + V (x)v = |v|^{p-2} v, v \in H^1 (\mathbb{R}^N ) $ where $N \ge 2$, $2 < p < 2^*$, $\epsilon> 0$ is a small parameter, and V is assumed to be bounded and bounded away from zero. When V has a local minimum point P, as $\epsilon \to 0$, we construct an infinite sequence of localized sign-changing solutions clustered at P and these solutions are of higher topological type in the sense that they are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. Our method combines the Byeon and Wang’s penalization approach and minimax method via a variant of the classical symmetric mountain pass theorem, and is rather robust without using any non-degeneracy conditions.

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