Global bifurcations of nodal solutions for coupled elliptic equations

TL;DR

This paper studies the global bifurcation structure of radial nodal solutions to a coupled elliptic system in a unit ball, revealing how solutions branch and differ based on the coupling parameter and nodal properties.

Contribution

It provides a comprehensive analysis of the bifurcation structure of nodal solutions for the coupled elliptic equations, including the characterization of solution branches and their disjointness for different nodal counts.

Findings

Identified exactly four solution curves for each nodal solution with k-1 zeros.

Established the disjointness of bifurcation structures for different nodal solutions.

Provided detailed nodal information for each bifurcating branch.

Abstract

We investigate the global bifurcation structure of the radial nodal solutions to the coupled elliptic equations \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u+u=u^3+\beta uv^2\mbox{ in }B_1 ,\nonumber -{\Delta}v+v=v^3+\beta u^2v\mbox{ in }B_1 ,\nonumber u,v\in H_{0,r}^1(B_1).\nonumber \end{array} \right. \end{equation} Here $B_1$ is a unit ball in $\mathbb{R}^3$ and $\beta\in\mathbb{R}$ the coupling constant is used as bifurcation parameter. For each $k$, the unique pair of nodal solutions $\pm w_k$ with exactly $k-1$ zeroes to the scalar field equation $-\Delta w + w=w^3$ generate exactly four synchronized solution curves and exactly four semi-trivial solution curves to the above system. We obtain a fairly complete global bifurcation structure of all bifurcating branches emanating from these eight solution curves of the system, and show that for different $k$ these bifurcation structures are disjoint. We obtain exact and distinct nodal information for each of the bifurcating branches, thus providing a fairly complete characterization of nodal solutions of the system in terms of the coupling.