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Bifurcations of elliptic systems with linear couplings. (English) Zbl 1428.35035

Summary: Consider the elliptic system with linearly coupled terms \[ \begin{cases} -\Delta u=\lambda v+f_1(u,v), & \text{ in }\Omega, \\ -\Delta v=\mu u+f_2(u,v), & \text{ in }\Omega, \\ u=0,v=0, & \text{ on }\partial\Omega, \end{cases} \] where \(\lambda, \mu \in \mathbb{R}\) are constants and \(\Omega \subset \mathbb{R}^N\) is a smooth bounded domain. We study the local and global bifurcations with respect to \(\mathcal{T}_0 := \{((\lambda, \mu),(0, 0)) \} \subset \mathbb{R}^2 \times X\), where \(X\) is a proper Banach space. Our results are of particular interest for obtaining nontrivial solutions in the case \(\lambda \neq \mu \).

MSC:

35B32 Bifurcations in context of PDEs
35J57 Boundary value problems for second-order elliptic systems
35J61 Semilinear elliptic equations
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