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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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