  We deal with the first eigenvalue for a system of two $p-$Laplacians with
Dirichlet and Neumann boundary conditions. If $\Delta_{p}w=\mbox{div}(|\nabla
w|^{p-2}w)$ stands for the $p-$Laplacian and
$\frac{\alpha}{p}+\frac{\beta}{q}=1,$ we consider $$ \begin{cases} -\Delta_pu=
\lambda \alpha |u|^{\alpha-2} u|v|^{\beta} &\text{ in }\Omega,\\ -\Delta_q v=
\lambda \beta |u|^{\alpha}|v|^{\beta-2}v &\text{ in }\Omega,\\ \end{cases} $$
with mixed boundary conditions $$ u=0, \qquad |\nabla v|^{q-2}\dfrac{\partial
v}{\partial \nu }=0, \qquad \text{on }\partial \Omega. $$ We show that there is
a first non trivial eigenvalue that can be characterized by the variational
minimization problem $$ \lambda_{p,q}^{\alpha,\beta} = \min
\left\{\dfrac{\displaystyle\int_{\Omega}\dfrac{|\nabla u|^p}{p}\, dx
+\int_{\Omega}\dfrac{|\nabla v|^q}{q}\, dx} {\displaystyle\int_{\Omega}
|u|^\alpha|v|^{\beta}\, dx} \colon (u,v)\in
\mathcal{A}_{p,q}^{\alpha,\beta}\right\}, $$ where $$
\mathcal{A}_{p,q}^{\alpha,\beta}=\left\{(u,v)\in W^{1,p}_0(\Omega)\times
W^{1,q}(\Omega)\colon uv\not\equiv0\text{ and
}\int_{\Omega}|u|^{\alpha}|v|^{\beta-2}v \, dx=0\right\}. $$ We also study the
limit of $\lambda_{p,q}^{\alpha,\beta} $ as $p,q\to \infty$ assuming that
$\frac{\alpha}{p} \to \Gamma \in (0,1)$, and $ \frac{q}{p} \to Q \in
(0,\infty)$ as $p,q\to \infty.$ We find that this limit problem interpolates
between the pure Dirichlet and Neumann cases for a single equation when we take
$Q=1$ and the limits $\Gamma \to 1$ and $\Gamma \to 0$.
