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 Delta(p)w = div(|del w|(p-2)del w) stands for the p-Laplacian and alpha/p + beta/q = 1, we consider {-Delta(p)u = lambda alpha vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar(beta) in Omega, -Delta(p)v = lambda beta vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2)v in Omega, with mixed boundary conditions u=0, vertical bar del v vertical bar(q-2) partial derivative v/partial derivative nu = 0, on partial derivative Omega. We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem lambda(alpha,beta)(p,q) - min {integral(Omega)vertical bar del u vertical bar(p)/p dx + integral(Omega)vertical bar del v vertical bar(q)/q dx/integral(Omega)vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta) dx :(u, v) is an element of A(p,q)(alpha,beta)}, where A(p,q)(alpha,beta) = {(u,v) is an element of W-0(1,p) (Omega) x W-1,W-q (Omega) : uv not equivalent to 0 and integral(Omega) vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2) v dx =0}. We also study the limit of lambda(alpha,beta)(p,q) as p, q -> infinity assuming that a/p -> Gamma is an element of (0, 1), and q/p -> Q is an element of (0,infinity) as p, q -> infinity. 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 -> 1 and Gamma -> 0. (c) 2015 Elsevier Ltd. All rights reserved.