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The first nontrivial eigenvalue for a system of $p-$Laplacians with Neumann and Dirichlet boundary conditions

We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If $Δ_{p}w=\mbox{div}(|\nabla w|^{p-2}w)$ stands for the $p-$Laplacian and $\fracα{p}+\fracβ{q}=1,$ we consider $$ \begin{cases} -Δ_pu= λα|u|^{α-2} u|v|^β &\text{ in }Ω,\\ -Δ_q v= λβ|u|^α|v|^{β-2}v &\text{ in }Ω,\\ \end{cases} $$ with mixed boundary conditions $$ u=0, \qquad |\nabla v|^{q-2}\dfrac{\partial v}{\partial ν}=0, \qquad \text{on }\partial Ω. $$ We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem $$ λ_{p,q}^{α,β} = \min \left\{\dfrac{\displaystyle\int_Ω\dfrac{|\nabla u|^p}{p}\, dx +\int_Ω\dfrac{|\nabla v|^q}{q}\, dx} {\displaystyle\int_Ω |u|^α|v|^β\, dx} \colon (u,v)\in \mathcal{A}_{p,q}^{α,β}\right\}, $$ where $$ \mathcal{A}_{p,q}^{α,β}=\left\{(u,v)\in W^{1,p}_0(Ω)\times W^{1,q}(Ω)\colon uv\not\equiv0\text{ and }\int_Ω|u|^α|v|^{β-2}v \, dx=0\right\}. $$ We also study the limit of $λ_{p,q}^{α,β} $ as $p,q\to \infty$ assuming that $\fracα{p} \to Γ\in (0,1)$, 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 $Γ\to 1$ and $Γ\to 0$.