Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global bifurcations of nodal solutions for coupled elliptic equations

Haoyu Li, Olímpio Hiroshi Miyagaki, Zhi-Qiang Wang

TL;DR

This work analyzes the global bifurcation structure of radial nodal solutions to a three-dimensional coupled cubic elliptic system on the unit ball, using the coupling constant $\beta$ as the bifurcation parameter. It identifies four synchronized and four semi-trivial branches for each nodal level $k$ and proves the existence of infinitely many global bifurcating branches from eight base curves, with distinct structures for different $k$. The approach combines a weighted eigenvalue problem, Morse-index calculations, Rabinowitz-type global bifurcation, and Liouville-type nonexistence theorems, complemented by careful nodal analysis and asymptotics as $\beta\to\infty$. The results yield a fairly complete nodal classification of the coupled system, reveal how coupling affects nodal patterns, and enable applications such as comparing scalar nodal solutions and refining nonexistence thresholds near $\beta=3$. Overall, the paper advances understanding of interspecies coupling effects in radial nodal solutions of coupled elliptic systems and provides a robust framework for analyzing similar problems.

Abstract

We investigate the global bifurcation structure of the radial nodal solutions to the coupled elliptic equations \begin{equation} \left\{ \begin{array}{lr} -Δu+u=u^3+βuv^2\mbox{ in }B_1 ,\nonumber -Δv+v=v^3+βu^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 $β\in\mathbb{R}$ 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 $-Δw + w=w^3$ 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.

Global bifurcations of nodal solutions for coupled elliptic equations

TL;DR

This work analyzes the global bifurcation structure of radial nodal solutions to a three-dimensional coupled cubic elliptic system on the unit ball, using the coupling constant as the bifurcation parameter. It identifies four synchronized and four semi-trivial branches for each nodal level and proves the existence of infinitely many global bifurcating branches from eight base curves, with distinct structures for different . The approach combines a weighted eigenvalue problem, Morse-index calculations, Rabinowitz-type global bifurcation, and Liouville-type nonexistence theorems, complemented by careful nodal analysis and asymptotics as . The results yield a fairly complete nodal classification of the coupled system, reveal how coupling affects nodal patterns, and enable applications such as comparing scalar nodal solutions and refining nonexistence thresholds near . Overall, the paper advances understanding of interspecies coupling effects in radial nodal solutions of coupled elliptic systems and provides a robust framework for analyzing similar problems.

Abstract

We investigate the global bifurcation structure of the radial nodal solutions to the coupled elliptic equations \begin{equation} \left\{ \begin{array}{lr} -Δu+u=u^3+βuv^2\mbox{ in }B_1 ,\nonumber -Δv+v=v^3+βu^2v\mbox{ in }B_1 ,\nonumber u,v\in H_{0,r}^1(B_1).\nonumber \end{array} \right. \end{equation} Here is a unit ball in and the coupling constant is used as bifurcation parameter. For each , the unique pair of nodal solutions with exactly zeroes to the scalar field equation 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 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.

Paper Structure

This paper contains 17 sections, 26 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Introduction
  3. Main results: Bifurcation theorems and a nonexistence result
  4. An application of Theorem \ref{['t:bifurcation']}
  5. Main results: Liouville type theorems
  6. Organization of this paper
  7. Proof of Theorems \ref{['t:bifurcation']} and \ref{['t:bifurcation2']}
  8. A weighted eigenvalue problem
  9. Determining the bifurcation parameters
  10. Proof of Assertion $(1)$ of Theorem \ref{['t:bifurcation']} and of Assertion $(1)$ of Theorem \ref{['t:bifurcation2']}
  11. Mappings from $\mathcal{T}^{1}_k$ to $\mathcal{ST}_k^l$ for $l=1,2,3,4$
  12. Proof of Assertions $(2-3)$ of Theorem \ref{['t:bifurcation']}
  13. Proofs of Assertions $(4-5)$ of Theorem \ref{['t:bifurcation']} and of Assertions $(2-5)$ of Theorem \ref{['t:bifurcation2']}
  14. Proof of Theorem \ref{['t:nonexistencebeta=3']}
  15. Proof of Theorem \ref{['t:Liouville']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that $k\in\mathbb{N}_+$. The bifurcation parameters of $\mathcal{T}^1_k$ form a decreasing sequence $\{\beta_{k,i}\}_{i=1}^\infty$ such that

Figures (1)

  • Figure 1: Bifurcations emanating from $\mathcal{T}^1_k$ and $\mathcal{ST}_k^1$

Theorems & Definitions (53)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.4
  • ...and 43 more