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

We deal with the first eigenvalue for a system of two pp-Laplacians with Dirichlet and Neumann boundary conditions. If Δpw=div(|∇w|p−2∇w)Δpw=div(|∇w|p−2∇w) stands for the pp-Laplacian and αp+βq=1, we consider {−Δpu=λα|u|α−2u|v|β  in  Ω,−Δqv=λβ|u|α|v|β−2v  in  Ω, with mixed boundary conditions u=0,|∇v|q−2∂v∂ν=0,on  ∂Ω. We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem λp,qα,β=min{∫Ω|∇u|ppdx+∫Ω|∇v|qqdx∫Ω|u|α|v|βdx:(u,v)∈Ap,qα,β}, where Ap,qα,β={(u,v)∈W01,p(Ω)×W1,q(Ω):uv≢0  and  ∫Ω|u|α|v|β−2vdx=0}. We also study the limit of λp,qα,β as p,q→∞p,q→∞ assuming that αp→Γ∈(0,1), and qp→Q∈(0,∞) as p,q→∞p,q→∞. We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take Q=1Q=1 and the limits Γ→1Γ→1 and Γ→0Γ→0.