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Register a new userMultiple nodal solutions having shared componentwise nodal numbers for coupled Schr{o}dinger equations
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We investigate the structure of nodal solutions for coupled nonlinear Schr{o}dinger 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} -{Delta}u_{j}+lambda u_{j}=mu u^{3}_{j}+sum_{i eq j}beta u_{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 $nleq 3$, $lambda>0$, $mu>0$, and $beta <0$. More precisely, let $p$ be a prime factor of $N$ and write $N=pB$. Suppose $betaleq-frac{mu}{p-1}$. 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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