The behavior of solutions to an elliptic equation involving a [formula omitted]-Laplacian and a [formula omitted]-Laplacian for large [formula omitted

Abstract

In this paper we study the behavior as p→∞ of solutions up,q to −Δpu−Δqu=0 in a bounded smooth domain Ω with a Lipschitz Dirichlet boundary datum u=g on ∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj→∞ such that upj,q→u∞ uniformly in Ω¯ and we prove that this limit u∞ is a solution to a variational problem, that, when the Lipschitz constant of the boundary datum is less than or equal to one, is given by the minimization of the Lq-norm of the gradient with a pointwise constraint on the gradient. In addition we show that the limit is a viscosity solution to a limit PDE problem that involves the q-Laplacian and the ∞-Laplacian.