Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations

Abstract

We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schrodinger equation $$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1(\mathbb {R}^N) \end{aligned}$$ where \(N\ge 2,\)\(2 0\) 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 \(\epsilon \rightarrow 0\), 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. It has been an open question whether the sign-changing solutions of higher topological type can be localized and our result gives an affirmative answer. The existing results in the literature have been subject to some geometrical or topological constraints that limit the number of localized sign-changing solutions. At a local minimum point of V, Bartsch et al. (Math Ann 338:147–185, 2007) proved the existence of N pairs of localized sign-changing solutions and D’Aprile and Pistoia (Ann Inst Henri Poincare Anal Non Lineaire 26:1423–1451, 2009) constructed 9 pairs of localized sign-changing solutions for \(N\ge 3\). Our result gives an unbounded sequence of such solutions. 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.