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Bifurcation of nonlinear elliptic system from the first eigenvalue. (English) Zbl 1103.35320
Summary: We study the following bifurcation problem in a bounded domain \(\Omega\) in \(\mathbb R^N\): \[ \begin{cases} -\Delta_p u=\lambda |u|^{\alpha}|v|^{\beta}v + f(x,u,v,\lambda)& \text{in } \Omega\\ -\Delta_q v=\lambda |u|^{\alpha}|v|^{\beta}u + g(x,u,v,\lambda) & \text{in } \Omega\\ (u,v)\in W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega). \end{cases} \] We prove that the principal eigenvalue \(\lambda_1\) of the following eigenvalue problem \[ \begin{cases} -\Delta_p u=\lambda |u|^{\alpha}|v|^{\beta}v & \text{in }\Omega\\ -\Delta_q v=\lambda |u|^{\alpha}|v|^{\beta}u & \text{in } \Omega\\ (u,v)\in W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega) \end{cases} \] is simple and isolated and we prove that \((\lambda_1,0,0)\) is a bifurcation point of the system mentioned above.
MSC:
35J50 Variational methods for elliptic systems
35B32 Bifurcations in context of PDEs
47J15 Abstract bifurcation theory involving nonlinear operators
47H11 Degree theory for nonlinear operators
47N20 Applications of operator theory to differential and integral equations
35J70 Degenerate elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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