We deal with the first eigenvalue for a system of two p-Laplacians with Dirichlet and Neumann boundary conditions. If Δpw=div(|∇w|p−2∇w) stands for the p-Laplacian and , we consider
with mixed boundary conditions
We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem
where
We also study the limit of as p,q→∞ assuming that , and as p,q→∞. 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 Γ→1 and Γ→0.