Multiple existence and qualitative property of nodal solutions for coupled elliptic equations

Abstract

The paper studies nodal solutions having prescribed componentwise nodal data for the following coupled nonlinear elliptic equations \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u_{j}+ u_{j}= u^{3}_{j}+\beta\sum_{i=1, i\neq j}^N u_{j}u_{i}^{2} \,\,\,\,\,\,\, \mbox{in}\ \Omega,\nonumber u_{j}\in H_{0,r}^{1}(\Omega), \,\,\,\,\,\,\,\,j=1,\dots,N.\nonumber \end{array} \right. \end{equation} Here, $\Omega\subset\mathbb{R}^n$ is a bounded and radial domain with $n=2,3$. The coupling constant $\beta\leq-1$ 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.