Bifurcations of nontrivial solutions of a cubic Helmholtz system

This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system
$\begin{equation*}
\begin{cases} -Δu - μu = \left( u^2 + b \: v^2 \right) u &\text{ on }
\mathbb{R}^3, \\ -Δv - νv = \left( v^2 + b \: u^2 \right) v &\text{ on
} \mathbb{R}^3. \end{cases} \end{equation*}$
It is shown that every point along any given branch of radial semitrivial solutions $(u_0, 0, b)$ 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 of solutions at infinity that are shown to decay like $\frac{1}{|x|}$ as $|x|\to\infty$.