| Summary: | We study the following bifurcation problem in a bounded domain $\Omega$ in $\mathbb{R}^N$:
$$\left\{\begin{array}{lll}
-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,+ f(x,u,v,\lambda)&
\mbox{in} \ \Omega\\
-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \, + g(x,u,v,\lambda) &
\mbox{in} \ \Omega\\
(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega). & \
\end{array}
\right.
$$
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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