Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations
Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations
复制标题
DOI：
10.1007/s00526-016-1094-4
复制
发表时间：
2016-12
影响因子：
2.1
通讯作者：
Shaowei Chen;Zhi-Qiang Wang
中科院分区：
文献类型：
--
作者：
Shaowei Chen;Zhi-Qiang Wang
研究了半经典非线性薛定谔方程$$\开始{aligned} -\Delta ^2\Delta v+V（x）v=| v| ^{p-2}v，\ v\in H^1（\mathbb {R}^N）\end{aligned}$$其中，是一个小参数，假设V是有界的，并且远离零有界。当V有局部极小点P时，构造了一个无穷序列的局部变号解，它们聚集在P上，并且这些解是由对称山路定理对高维对称链接结构的极大极小刻画得到的，在这个意义上它们是高维拓扑型的.高阶拓扑型变号解能否局部化一直是一个悬而未决的问题，我们的结果给出了肯定的答案。文献中现有的结果受到一些几何或拓扑约束的限制，这些约束限制了局部变号解的数量。在V的局部极小点处，Bartsch等人（Math Ann 338：147-185，2007）证明了N对局部变号解的存在性，并且D 'Aprile和佩斯托亚（Ann Inst Hénri Poincare Anal Non Linéaire 26：1423-1451，2009）构造了V的9对局部变号解。我们的结果给出了一个无界序列的这种解决方案。我们的方法结合了Byeon和Wang的惩罚方法和Minimax方法，通过经典对称山路定理的变体，并且在不使用任何非退化条件的情况下是相当稳健的。
We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schrödinger equation $$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1(\mathbb {R}^N) \end{aligned}$$where,is a small parameter, andVis assumed to be bounded and bounded away from zero. WhenVhas a local minimum pointP, as, we construct an infinite sequence of localized sign-changing solutions clustered atPand 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 ofV, Bartsch et al. (Math Ann 338:147–185, 2007) proved the existence ofNpairs of localized sign-changing solutions and D’Aprile and Pistoia (Ann Inst Hénri Poincare Anal Non Linéaire 26:1423–1451, 2009) constructed 9 pairs of localized sign-changing solutions for. 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.