Bifurcations of nontrivial solutions of a cubic Helmholtz system

Abstract

Abstract This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u - \mu u = \left( u^2 + b \: v^2 \right) u &\text{ on } \mathbb{R}^3, \\ -{\it\Delta} v - \nu v = \left( v^2 + b \: u^2 \right) v &\text{ on } \mathbb{R}^3. \end{cases} \end{array}$$ It is shown that every point along any given branch of radial semitrivial solutions (u0, 0, b) or diagonal solutions (ub, ub, b) (for μ = ν) is a bifurcation point. Our analysis is based on a detailed investigation of the oscillatory behavior and the decay of solutions at infinity.