# Bifurcations of nontrivial solutions of a cubic Helmholtz system

Mandel, Rainer; Scheider, Dominic

Abstract:
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 $|x|\to\infty$.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X urn:nbn:de:swb:90-873141 KITopen-ID: 1000087314 Verlag KIT, Karlsruhe Umfang 31 S. Serie CRC 1173 ; 2018/32 Schlagworte nonlinear Helmholtz sytem, bifurcation
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