arXiv 2005.138602020-05math.AP

耦合薛定谔方程具有共享分量节点数的多节点解

Multiple nodal solutions having shared componentwise nodal numbers for coupled Schrödinger equations

Haoyu Li, Zhi-Qiang Wang

备注：: 39 pages

摘要：

本文研究了排斥耦合区域中耦合非线性薛定谔方程节点解的结构。在其他的结果中，对于下面的耦合系统的$N$方程，我们证明了存在无穷多个节点的解决方案，共享相同的组件指定的节点数\begin{equation}\label{ab} \left\{ \begin{array}{lr} -Δu_{j}+λu_{j}=μu^{3}_{j}+\sum_{i\neq j}βu_{j}u_{i}^{2} \,\,\,\,\,\,\, in\ \W , u_{j}\in H_{0,r}^{1}(\W), \,\,\,\,\,\,\,\,j=1,\dots,N, \end{array} \right. \end{equation}其中$\W$是一个径向域在$\mathbb R^n$的$n\leq 3$，$λ>0$，$μ>0$，和$β<0$。更准确地说，让$p$是$N$的素因子，并写为$N=pB$。假设为$β\leq-\fracμ{p-1}$。则对于任何给定的非负整数$P_{1},P_{2},\dots,P_{B}$，（\ref{ab}）有无穷多个解$(u_{1},\dots,u_{N})$，使得这些解中的每一个都满足相同的性质：对于$b=1,...,B$，对于$i=1,...,p$，$u_{pb-p+i}$精确地改变符号$P_b$次.结果揭示了由于溶液组分间的分离而导致的排斥耦合体系中溶液结构的复杂性。我们的方法是结合一个热流的方法作为变形的对称山口定理的极小极大建设使用$\mathbb Z_p$组作用指数。我们的方法是强大的，也允许给一个解决方案的存在，而不假设任何对称的耦合。

英文摘要：

We investigate the structure of nodal solutions for coupled nonlinear Schrödinger equations in the repulsive coupling regime. Among other results, for the following coupled system of $N$ equations, we prove the existence of infinitely many nodal solutions which share the same componentwise-prescribed nodal numbers \begin{equation}\label{ab} \left\{ \begin{array}{lr} -Δu_{j}+λu_{j}=μu^{3}_{j}+\sum_{i\neq j}βu_{j}u_{i}^{2} \,\,\,\,\,\,\, in\ \W , u_{j}\in H_{0,r}^{1}(\W), \,\,\,\,\,\,\,\,j=1,\dots,N, \end{array} \right. \end{equation} where $\W$ is a radial domain in $\mathbb R^n$ for $n\leq 3$, $λ>0$, $μ>0$, and $β<0$. More precisely, let $p$ be a prime factor of $N$ and write $N=pB$. Suppose $β\leq-\fracμ{p-1}$. Then for any given non-negative integers $P_{1},P_{2},\dots,P_{B}$, (\ref{ab}) has infinitely many solutions $(u_{1},\dots,u_{N})$ such that each of these solutions satisfies the same property: for $b=1,...,B$, $u_{pb-p+i}$ changes sign precisely $P_b$ times for $i=1,...,p$. The result reveals the complex nature of the solution structure in the repulsive coupling regime due to componentwise segregation of solutions. Our method is to combine a heat flow approach as deformation with a minimax construction of the symmetric mountain pass theorem using a $\mathbb Z_p$ group action index. Our method is robust, also allowing to give the existence of one solution without assuming any symmetry of the coupling.

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