Multiple existence and qualitative property of nodal solutions for coupled elliptic equations
Haoyu Li, Zhi-Qiang Wang
TL;DR
The paper addresses a repulsively coupled N-component elliptic system on bounded radial domains in dimensions two and three, proving the existence of infinitely many vector solutions with componentwise prescribed nodal data and providing sharp bounds on inter-component nodal differences. It develops a parabolic-flow–based symmetric mountain-pass framework, embedding the variational problem into a genus-counting strategy with invariant sets to produce nodal solutions. The results cover both positive and nodal solutions, offering detailed inter-component nodal estimates and demonstrating nonexistence phenomena under the repulsive regime. This work extends previous radial analyses by introducing a unified mechanism for prescribing nodal data and controlling nodal interactions across multiple components, with potential implications for the qualitative understanding of multi-component Bose-Einstein condensate models and standing-wave solutions of coupled nonlinear Schrödinger systems.
Abstract
The paper studies nodal solutions having prescribed componentwise nodal data for the following coupled nonlinear elliptic equations \begin{equation} \left\{ \begin{array}{lr} -Δu_{j}+ u_{j}= u^{3}_{j}+β\sum_{i=1, i\neq j}^N u_{j}u_{i}^{2} \,\,\,\,\,\,\, \mbox{in}\ Ω,\nonumber u_{j}\in H_{0,r}^{1}(Ω), \,\,\,\,\,\,\,\,j=1,\dots,N.\nonumber \end{array} \right. \end{equation} Here, $Ω\subset\mathbb{R}^n$ is a bounded and radial domain with $n=2,3$. The coupling constant $β\leq-1$ is in the repulsive regime. We investigate the solution structure for both positive and nodal solutions, proving multiple existence of solutions with prescribed nodal data and providing qualitative estimates for the nodal numbers of the inter-componentwise differences of solutions with both upper and lower bounds. Our general framework is for nodal solutions though our results are new also for positive solutions.