The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p

In this paper we study the behavior as p -> infinity of solutions u(p,q) to -Delta(p)u - Delta(q)u = 0 in a bounded smooth domain Omega with a Lipschitz Dirichlet boundary datum u = g on partial derivative Omega. We find that there is a uniform limit of a subsequence of solutions, that is, there is p(j) -> infinity such that u(pi,q) -> u(infinity) uniformly in (Omega) over bar and we prove that this limit u(infinity) 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 L-q-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 infinity-Laplacian. (C) 2016 Elsevier Ltd. All rights reserved.