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Bifurcation results of positive solutions for an indefinite nonlinear elliptic system. II. (English) Zbl 1277.35162
Summary: We study local and global bifurcations of the following elliptic system \[ \begin{cases} -\Delta u-au=\mu_1u^3+\beta uv^2,\quad &\text{in }\Omega\\-\Delta v-av=\mu_2v^3+\beta vu^2,\quad &\text{in }\Omega\\u,v>0\text{ in }\Omega,\, u=v=0,\quad &\text{on }\partial\Omega,\end{cases} \] where the constant \(a\) is greater than the principal eigenvalue of \((-\Delta,\Omega)\), and \(\Omega\subset\mathbb R^N\) is a bounded smooth domain \((N\leq 3)\). The parameters \(\mu_1,\mu_2\) are real numbers. We identify some intervals of \(\beta\) in which there exist finitely many bifurcation points with respect to a trivial solution branch. In particular, when \(N=1\) or \(\Omega\) is radial, every bifurcation branch exists globally and is characterized by a nodal property for the difference of the two components of the solutions.
For Part I of this paper see [Discrete Contin. Dyn. Syst. 33, No. 1, 335–344 (2013; Zbl 06146691)].

35J47 Second-order elliptic systems
35B32 Bifurcations in context of PDEs
35B09 Positive solutions to PDEs
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