论文标题
多个节点解决方案，具有共享的componentwise节点数字的耦合schrödinger方程
Multiple nodal solutions having shared componentwise nodal numbers for coupled Schrödinger equations
论文作者
论文摘要
我们研究了排斥耦合方程中耦合非线性schrödinger方程的节点溶液的结构。除其他结果外，对于以下$ n $方程的耦合系统，我们证明存在无限的许多节点解决方案，这些解决方案共享相同的componentWise probrecreped nodal numbers \ begin \ begin {equation} \ label {ab} \左边\{ \ begin {array} {lr} -ΔU_{j}+λu_{J} =μU^{3} _ {J}+\ sum_ {i \ neq j}βu_{j} u_ {j} u_ {i}^{i}^{2}^{2} \，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\， u_ {j} \ in H_ {0，r}^{1}（\ w），\，\，\，\，\，\，\，\，\，\，j = 1，j = 1，\ dot \ end {array} \正确的。 \ end {equation}其中$ \ w $是$ \ mathbb r^n $ in $ n \ leq 3 $，$λ> 0 $，$μ> 0 $和$β<0 $的径向域。更确切地说，让$ p $是$ n $的主要因素，并写入$ n = pb $。假设$β\ leq- \fracμ{p-1} $。然后，对于任何给定的非阴性整数$ p_ {1}，p_ {2}，\ dots，p_ {b} $，（\ ref ref {ab}）具有无限的许多解决方案$（u_ {1}，\ dots，\ dots，dots，d dots，u_ {n}）$，使这些解决方案都满足这些属性的属性。 $ u_ {pb-p+i} $更改标志$ i = 1，...，p $。结果揭示了由于解决方案的组分隔离而导致的排斥耦合方案中溶液结构的复杂性质。我们的方法是使用$ \ Mathbb Z_P $组动作索引将热流接入作为变形与对称山通定理的最小值结构相结合。我们的方法是强大的，也允许在不假设耦合的任何对称性的情况下给出一个溶液的存在。
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.
