Bifurcation of nonlinear elliptic system from the first eigenvalue

A. El Khalil, 3-1390, Boul. Décaire Montreal (Qc) H4L 3N1, Canada
M. Ouanan, Dhar-Mahraz, Atlas-Fes, Fes, Morocco
A. Touzani, Dhar-Mahraz, Atlas-Fes, Fes, Morocco

E. J. Qualitative Theory of Diff. Equ., No. 21. (2003), pp. 1-18.

Abstract: We study the following bifurcation problem in a bounded domain $\Omega$ in $\RR^N$:
$$\left\{\begin{array}{lll}
-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,+ f(x,u,v,\lambda)&
\mbox{in} \ \Omega\\
-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \, + g(x,u,v,\lambda) &
\mbox{in} \ \Omega\\
(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega). & \
\end{array}
\right.
$$
We prove that the principal eigenvalue $\lambda_1$ of the following eigenvalue problem
$$\left\{\begin{array}{lll}
-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,& \mbox{in} \ \Omega\\
-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \,& \mbox{in} \ \Omega\\
(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)& \
\end{array}
\right.$$
is simple and isolated and we prove that $(\lambda_1,0,0)$ is a bifurcation point of the system mentioned above.

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