The limit as p → ∞ in the eigenvalue problem for a system of p-Laplacians

In this paper, we study the behavior as p -> infinity of eigenvalues and eigenfunctions of a system of p-Laplacians, that is {-Delta(p)u = lambda alpha u(alpha-1)v(beta) Omega, -Delta(p)u =lambda alpha u(alpha) v(beta-1) Omega, u = v = 0, partial derivative Omega, in a bounded smooth domain Omega. Here alpha + beta = p. We assume that alpha/p -> Gamma and beta/p -> 1 -Gamma as p -> infinity and we prove that for the first eigenvalue lambda(1, p) we have (lambda(1, p))(1/ p) -> lambda(infinity) = 1/max(x is an element of Omega)dist(x, partial derivative Omega) Concerning the eigenfunctions (u(p), v(p)) associated with lambda(1, p) normalized by integral(Omega)vertical bar u(p)vertical bar(alpha)vertical bar vp vertical bar(beta) = 1, there is a uniform limit (u(infinity), v(infinity)) that is a solution to a limit minimization problem as well as a viscosity solution to {min{-Delta(infinity)u(infinity), vertical bar Delta u(infinity)vertical bar - lambda(infinity)u(infinity)(Gamma)v(infinity)(1-Gamma)} = 0, min{-Delta(infinity)v(infinity), vertical bar Delta u(infinity)vertical bar - lambda(infinity)u(infinity)(Gamma)v(infinity)(1-Gamma)} = 0 In addition, we also analyze the limit PDE when we consider higher eigenvalues.